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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 738520, 41 pages
http://dx.doi.org/10.1155/2011/738520
Research Article

Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic

Received 8 October 2010; Accepted 3 January 2011

Academic Editor: Miroslavaย Růžičková

Copyright ยฉ 2011 Roman Šimon Hilscher and Petr Zemánek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce the ๐‘€ ( ๐œ† ) -function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.

1. Introduction

In this paper we develop systematically the Weyl-Titchmarsh theory for time scale symplectic systems. Such systems unify and extend the classical linear Hamiltonian differential systems and discrete symplectic and Hamiltonian systems, including the Sturm-Liouville differential and difference equations of arbitrary even order. As the research in the Weyl-Titchmarsh theory has been very active in the last years, we contribute to this development by presenting a theory which directly generalizes and unifies the results in several recent papers, such as [14] and partly in [514].

Historically, the theory nowadays called by Weyl and Titchmarsh started in [15] by the investigation of the second-order linear differential equation ๎€ท ๐‘Ÿ ( ๐‘ก ) ๐‘ง ๎…ž ๎€ธ [ ( ๐‘ก ) + ๐‘ž ( ๐‘ก ) ๐‘ง ( ๐‘ก ) = ๐œ† ๐‘ง ( ๐‘ก ) , ๐‘ก โˆˆ 0 , โˆž ) , ( 1 . 1 ) where ๐‘Ÿ , ๐‘ž โˆถ [ 0 , โˆž ) โ†’ โ„ are continuous, ๐‘Ÿ ( ๐‘ก ) > 0 , and ๐œ† โˆˆ โ„‚ , is a spectral parameter. By using a geometrical approach it was showed that (1.1) can be divided into two classes called the limit circle and limit point meaning that either all solutions of (1.1) are square integrable for all ๐œ† โˆˆ โ„‚ โงต โ„ or there is a unique (up to a multiplicative constant) square-integrable solution of (1.1) on [ 0 , โˆž ) . Analytic methods for the investigation of (1.1) have been introduced in a series of papers starting with [16]; see also [17]. We refer to [1820] for an overview of the original contributions to the Weyl-Titchmarsh theory for (1.1); see also [21]. Extensions of the Weyl-Titchmarsh theory to more general equations, namely, to the linear Hamiltonian differential systems ๐‘ง ๎…ž [ ] ๐‘ง [ ( ๐‘ก ) = ๐œ† ๐ด ( ๐‘ก ) + ๐ต ( ๐‘ก ) ( ๐‘ก ) , ๐‘ก โˆˆ 0 , โˆž ) , ( 1 . 2 ) was initiated in [22] and developed further in [6, 8, 10, 11, 2338].

According to [19], the first paper dealing with the parallel discrete time Weyl theory for second-order difference equations appears to be the work mentioned in [39]. Since then a long time elapsed until the theory of difference equations attracted more attention. The Weyl-Titchmarsh theory for the second-order Sturm-Liouville difference equations was developed in [22, 40, 41]; see also the references in [19]. For higher-order Sturm-Liouville difference equations and linear Hamiltonian difference systems, such as ฮ” ๐‘ฅ ๐‘˜ = ๐ด ๐‘˜ ๐‘ฅ ๐‘˜ + 1 + ๎‚€ ๐ต ๐‘˜ + ๐œ† ๐‘Š ๐‘˜ [ 2 ] ๎‚ ๐‘ข ๐‘˜ , ฮ” ๐‘ข ๐‘˜ = ๎‚€ ๐ถ ๐‘˜ โˆ’ ๐œ† ๐‘Š ๐‘˜ [ 1 ] ๎‚ ๐‘ฅ ๐‘˜ + 1 โˆ’ ๐ด โˆ— ๐‘˜ ๐‘ข ๐‘˜ [ , ๐‘˜ โˆˆ 0 , โˆž ) โ„ค , ( 1 . 3 ) where ๐ด ๐‘˜ , ๐ต ๐‘˜ , ๐ถ ๐‘˜ , ๐‘Š ๐‘˜ [ 1 ] , ๐‘Š ๐‘˜ [ 2 ] are complex ๐‘› ร— ๐‘› matrices such that ๐ต ๐‘˜ and ๐ถ ๐‘˜ are Hermitian and ๐‘Š ๐‘˜ [ 1 ] and ๐‘Š ๐‘˜ [ 2 ] are Hermitian and nonnegative definite, the Weyl-Titchmarsh theory was studied in [9, 14, 42]. Recently, the results for linear Hamiltonian difference systems were generalized in [1, 2] to discrete symplectic systems ๐‘ฅ ๐‘˜ + 1 = ๐’œ ๐‘˜ ๐‘ฅ ๐‘˜ + โ„ฌ ๐‘˜ ๐‘ข ๐‘˜ , ๐‘ข ๐‘˜ + 1 = ๐’ž ๐‘˜ ๐‘ฅ ๐‘˜ + ๐’Ÿ ๐‘˜ ๐‘ข ๐‘˜ + ๐œ† ๐’ฒ ๐‘˜ ๐‘ฅ ๐‘˜ + 1 [ , ๐‘˜ โˆˆ 0 , โˆž ) โ„ค , ( 1 . 4 ) where ๐’œ ๐‘˜ , โ„ฌ ๐‘˜ , ๐’ž ๐‘˜ , ๐’Ÿ ๐‘˜ , ๐’ฒ ๐‘˜ are complex ๐‘› ร— ๐‘› matrices such that ๐’ฒ ๐‘˜ is Hermitian and nonnegative definite and the 2 ๐‘› ร— 2 ๐‘› transition matrix in (1.4) is symplectic, that is, ๐’ฎ ๐‘˜ ๎‚ต ๐’œ โˆถ = ๐‘˜ โ„ฌ ๐‘˜ ๐’ž ๐‘˜ ๐’Ÿ ๐‘˜ ๎‚ถ , ๐’ฎ โˆ— ๐‘˜ ๐’ฅ ๐’ฎ ๐‘˜ ๎‚ต ๎‚ถ = ๐’ฅ , ๐’ฅ โˆถ = 0 ๐ผ โˆ’ ๐ผ 0 . ( 1 . 5 )

In the unifying theory for differential and difference equations—the theory of time scales—the classification of second-order Sturm-Liouville dynamic equations ๐‘ฆ ฮ” ฮ” ( ๐‘ก ) + ๐‘ž ( ๐‘ก ) ๐‘ฆ ๐œŽ ( ๐‘ก ) = ๐œ† ๐‘ฆ ๐œŽ ( [ ๐‘ก ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 1 . 6 ) to be of the limit point or limit circle type is given in [4, 43]. These two papers seem to be the only ones on time scales which are devoted to the Weyl-Titchmarsh theory for the second order dynamic equations. Another way of generalizing the Weyl-Titchmarsh theory for continuous and discrete Hamiltonian systems was presented in [3, 5]. In these references the authors consider the linear Hamiltonian system ๐‘ฅ ฮ” ( ๐‘ก ) = ๐ด ( ๐‘ก ) ๐‘ฅ ๐œŽ ( ๎€บ ๐‘ก ) + ๐ต ( ๐‘ก ) + ๐œ† ๐‘Š 2 ( ๎€ป ๐‘ข ๐‘ก ) ๐‘ข ( ๐‘ก ) , ฮ” ( ๎€บ ๐‘ก ) = ๐ถ ( ๐‘ก ) โˆ’ ๐œ† ๐‘Š 1 ( ๎€ป ๐‘ฅ ๐‘ก ) ๐œŽ ( ๐‘ก ) โˆ’ ๐ด โˆ— ( [ ๐‘ก ) ๐‘ข ( ๐‘ก ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 1 . 7 ) on the so-called Sturmian or general time scales, respectively. Here ๐‘“ ฮ” ( ๐‘ก ) is the time scale ฮ” -derivative and ๐‘“ ๐œŽ ( ๐‘ก ) โˆถ = ๐‘“ ( ๐œŽ ( ๐‘ก ) ) , where ๐œŽ ( ๐‘ก ) is the forward jump at ๐‘ก ; see the time scale notation in Section 2.

In the present paper we develop the Weyl-Titchmarsh theory for more general linear dynamic systems, namely, the time scale symplectic systems ๐‘ฅ ฮ” ( ๐‘ข ๐‘ก ) = ๐’œ ( ๐‘ก ) ๐‘ฅ ( ๐‘ก ) + โ„ฌ ( ๐‘ก ) ๐‘ข ( ๐‘ก ) , ฮ” ( ๐‘ก ) = ๐’ž ( ๐‘ก ) ๐‘ฅ ( ๐‘ก ) + ๐’Ÿ ( ๐‘ก ) ๐‘ข ( ๐‘ก ) โˆ’ ๐œ† ๐’ฒ ( ๐‘ก ) ๐‘ฅ ๐œŽ [ ( ๐‘ก ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( ๐’ฎ ๐œ† ) where ๐’œ , โ„ฌ , ๐’ž , ๐’Ÿ , ๐’ฒ are complex ๐‘› ร— ๐‘› matrix functions on [ ๐‘Ž , โˆž ) ๐•‹ , ๐’ฒ ( ๐‘ก ) is Hermitian and nonnegative definite, ๐œ† โˆˆ โ„‚ , and the 2 ๐‘› ร— 2 ๐‘› coefficient matrix in system ( ๐’ฎ ๐œ† ) satisfies ๎‚ต ๎‚ถ ๐’ฎ ( ๐‘ก ) โˆถ = ๐’œ ( ๐‘ก ) โ„ฌ ( ๐‘ก ) ๐’ž ( ๐‘ก ) ๐’Ÿ ( ๐‘ก ) , ๐’ฎ โˆ— ( ๐‘ก ) ๐’ฅ + ๐’ฅ ๐’ฎ ( ๐‘ก ) + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— [ ( ๐‘ก ) ๐’ฅ ๐’ฎ ( ๐‘ก ) = 0 , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 1 . 8 ) where ๐œ‡ ( ๐‘ก ) โˆถ = ๐œŽ ( ๐‘ก ) โˆ’ ๐‘ก is the graininess of the time scale. The spectral parameter ๐œ† is only in the second equation of system ( ๐’ฎ ๐œ† ). This system was introduced in [44], and it naturally unifies the previously mentioned continuous, discrete, and time scale linear Hamiltonian systems (having the spectral parameter in the second equation only) and discrete symplectic systems into one framework. Our main results are the properties of the ๐‘€ ( ๐œ† ) function, the geometric description of the Weyl disks, and characterizations of the limit point and limit circle cases for the time scale symplectic system ( ๐’ฎ ๐œ† ). In addition, we give a formula for the ๐ฟ 2 ๐’ฒ solutions of a nonhomogeneous time scale symplectic system in terms of its Green function. These results generalize and unify in particular all the results in [14] and some results from [514]. The theory of time scale symplectic systems or Hamiltonian systems is a topic with active research in recent years; see, for example, [4451]. This paper can be regarded not only as a completion of these papers by establishing the Weyl-Titchmarsh theory for time scale symplectic systems but also as a comparison of the corresponding continuous and discrete time results. The references to particular statements in the literature are displayed throughout the text. Many results of this paper are new even for (1.6), being a special case of system ( ๐’ฎ ๐œ† ). An overview of these new results for (1.6) will be presented in our subsequent work.

This paper is organized as follows. In the next section we recall some basic notions from the theory of time scales and linear algebra. In Section 3 we present fundamental properties of time scale symplectic systems with complex coefficients, including the important Lagrange identity (Theorem 3.5) and other formulas involving their solutions. In Section 4 we define the time scale ๐‘€ ( ๐œ† ) -function for system ( ๐’ฎ ๐œ† ) and establish its basic properties in the case of the regular spectral problem. In Section 5 we introduce the Weyl disks and circles for system ( ๐’ฎ ๐œ† ) and describe their geometric structure in terms of contractive matrices in โ„‚ ๐‘› ร— ๐‘› . The properties of the limiting Weyl disk and Weyl circle are then studied in Section 6, where we also prove that system ( ๐’ฎ ๐œ† ) has at least ๐‘› linearly independent solutions in the space ๐ฟ 2 ๐’ฒ (see Theorem 6.7). In Section 7 we define the system ( ๐’ฎ ๐œ† ) to be in the limit point and limit circle case and prove several characterizations of these properties. In the final section we consider the system ( ๐’ฎ ๐œ† ) with a nonhomogeneous term. We construct its Green function, discuss its properties, and characterize the ๐ฟ 2 ๐’ฒ solutions of this nonhomogeneous system in terms of the Green function (Theorem 8.5). A certain uniqueness result is also proven for the limit point case.

2. Time Scales

Following [52, 53], a time scale ๐•‹ is any nonempty and closed subset of โ„ . A bounded time scale can be therefore identified as [ ๐‘Ž , ๐‘ ] ๐•‹ โˆถ = [ ๐‘Ž , ๐‘ ] โˆฉ ๐•‹ which we call the time scale interval, where ๐‘Ž โˆถ = m i n ๐•‹ and ๐‘ โˆถ = m a x ๐•‹ . Similarly, a time scale which is unbounded above has the form [ ๐‘Ž , โˆž ) ๐•‹ โˆถ = [ ๐‘Ž , โˆž ) โˆฉ ๐•‹ . The forward and backward jump operators on a time scale are denoted by ๐œŽ ( ๐‘ก ) and ๐œŒ ( ๐‘ก ) and the graininess function by ๐œ‡ ( ๐‘ก ) โˆถ = ๐œŽ ( ๐‘ก ) โˆ’ ๐‘ก . If not otherwise stated, all functions in this paper are considered to be complex valued. A function ๐‘“ on [ ๐‘Ž , ๐‘ ] ๐•‹ is called piecewise rd-continuous; we write ๐‘“ โˆˆ C p r d on [ ๐‘Ž , ๐‘ ] ๐•‹ if the right-hand limit ๐‘“ ( ๐‘ก + ) exists finite at all right-dense points ๐‘ก โˆˆ [ ๐‘Ž , ๐‘ ) ๐•‹ , and the left-hand limit ๐‘“ ( ๐‘ก โˆ’ ) exists finite at all left-dense points ๐‘ก โˆˆ ( ๐‘Ž , ๐‘ ] ๐•‹ and ๐‘“ is continuous in the topology of the given time scale at all but possibly finitely many right-dense points ๐‘ก โˆˆ [ ๐‘Ž , ๐‘ ) ๐•‹ . A function ๐‘“ on [ ๐‘Ž , โˆž ) ๐•‹ is piecewise rd-continuous; we write ๐‘“ โˆˆ C p r d on [ ๐‘Ž , โˆž ) ๐•‹ if ๐‘“ โˆˆ C p r d on [ ๐‘Ž , ๐‘ ] ๐•‹ for every ๐‘ โˆˆ ( ๐‘Ž , โˆž ) ๐•‹ . An ๐‘› ร— ๐‘› matrix-valued function ๐‘“ is called regressive on a given time scale interval if ๐ผ + ๐œ‡ ( ๐‘ก ) ๐‘“ ( ๐‘ก ) is invertible for all ๐‘ก in this interval.

The time scale ฮ” -derivative of a function ๐‘“ at a point ๐‘ก is denoted by ๐‘“ ฮ” ( ๐‘ก ) ; see [52, Definition  1.10]. Whenever ๐‘“ ฮ” ( ๐‘ก ) exists, the formula ๐‘“ ๐œŽ ( ๐‘ก ) = ๐‘“ ( ๐‘ก ) + ๐œ‡ ( ๐‘ก ) ๐‘“ ฮ” ( ๐‘ก ) holds true. The product rule for the ฮ” -differentiation of the product of two functions has the form ( ๐‘“ ๐‘” ) ฮ” ( ๐‘ก ) = ๐‘“ ฮ” ( ๐‘ก ) ๐‘” ( ๐‘ก ) + ๐‘“ ๐œŽ ( ๐‘ก ) ๐‘” ฮ” ( ๐‘ก ) = ๐‘“ ฮ” ( ๐‘ก ) ๐‘” ๐œŽ ( ๐‘ก ) + ๐‘“ ( ๐‘ก ) ๐‘” ฮ” ( ๐‘ก ) . ( 2 . 1 ) A function ๐‘“ on [ ๐‘Ž , ๐‘ ] ๐•‹ is called piecewise rd-continuously ฮ” -differentiable; we write ๐‘“ โˆˆ C 1 p r d on [ ๐‘Ž , ๐‘ ] ๐•‹ ; if it is continuous on [ ๐‘Ž , ๐‘ ] ๐•‹ , then ๐‘“ ฮ” ( ๐‘ก ) exists at all except for possibly finitely many points ๐‘ก โˆˆ [ ๐‘Ž , ๐œŒ ( ๐‘ ) ] ๐•‹ , and ๐‘“ ฮ” โˆˆ C p r d on [ ๐‘Ž , ๐œŒ ( ๐‘ ) ] ๐•‹ . As a consequence we have that the finitely many points ๐‘ก ๐‘– at which ๐‘“ ฮ” ( ๐‘ก ๐‘– ) does not exist belong to ( ๐‘Ž , ๐‘ ) ๐•‹ and these points ๐‘ก ๐‘– are necessarily right-dense and left-dense at the same time. Also, since at those points we know that ๐‘“ ฮ” ( ๐‘ก + ๐‘– ) and ๐‘“ ฮ” ( ๐‘ก โˆ’ ๐‘– ) exist finite, we replace the quantity ๐‘“ ฮ” ( ๐‘ก ๐‘– ) by ๐‘“ ฮ” ( ๐‘ก ยฑ ๐‘– ) in any formula involving ๐‘“ ฮ” ( ๐‘ก ) for all ๐‘ก โˆˆ [ ๐‘Ž , ๐œŒ ( ๐‘ ) ] ๐•‹ . Similarly as above we define ๐‘“ โˆˆ C 1 p r d on [ ๐‘Ž , โˆž ) ๐•‹ . The time scale integral of a piecewise rd-continuous function ๐‘“ over [ ๐‘Ž , ๐‘ ] ๐•‹ is denoted by โˆซ ๐‘ ๐‘Ž ๐‘“ ( ๐‘ก ) ฮ” ๐‘ก and over [ ๐‘Ž , โˆž ) ๐•‹ by โˆซ โˆž ๐‘Ž ๐‘“ ( ๐‘ก ) ฮ” ๐‘ก provided this integral is convergent in the usual sense; see [52, Definitions  1.71 and  1.82].

Remark 2.1. As it is known in [52, Theorem  5.8] and discussed in [54, Remark  3.8], for a fixed ๐‘ก 0 โˆˆ [ ๐‘Ž , ๐‘ ] ๐•‹ and a piecewise rd-continuous ๐‘› ร— ๐‘› matrix function ๐ด ( โ‹… ) on [ ๐‘Ž , ๐‘ ] ๐•‹ which is regressive on [ ๐‘Ž , ๐‘ก 0 ) ๐•‹ , the initial value problem ๐‘ฆ ฮ” ( ๐‘ก ) = ๐ด ( ๐‘ก ) ๐‘ฆ ( ๐‘ก ) for ๐‘ก โˆˆ [ ๐‘Ž , ๐œŒ ( ๐‘ ) ] ๐•‹ with ๐‘ฆ ( ๐‘ก 0 ) = ๐‘ฆ 0 has a unique solution ๐‘ฆ ( โ‹… ) โˆˆ C 1 p r d on [ ๐‘Ž , ๐‘ ] ๐•‹ for any ๐‘ฆ 0 โˆˆ โ„‚ ๐‘› . Similarly, this result holds on [ ๐‘Ž , โˆž ) ๐•‹ .

Let us recall some matrix notations from linear algebra used in this paper. Given a complex square matrix ๐‘€ , by ๐‘€ โˆ— , ๐‘€ > 0 , ๐‘€ โ‰ฅ 0 , ๐‘€ < 0 , ๐‘€ โ‰ค 0 , r a n k ๐‘€ , K e r ๐‘€ , d e f ๐‘€ , we denote, respectively, the conjugate transpose, positive definiteness, positive semidefiniteness, negative definiteness, negative semidefiniteness, rank, kernel, and the defect (i.e., the dimension of the kernel) of the matrix ๐‘€ . Moreover, we will use the notation I m ( ๐‘€ ) โˆถ = ( ๐‘€ โˆ’ ๐‘€ โˆ— ) / ( 2 ๐‘– ) and R e ( ๐‘€ ) โˆถ = ( ๐‘€ + ๐‘€ โˆ— ) / 2 for the Hermitian components of the matrix ๐‘€ ; see [55, pages  268-269] or [56, Fact  3.5.24]. This notation will be also used with ๐œ† โˆˆ โ„‚ , and in this case I m ( ๐œ† ) and R e ( ๐œ† ) represent the imaginary and real parts of ๐œ† .

Remark 2.2. If the matrix I m ( ๐‘€ ) is positive or negative definite, then the matrix ๐‘€ is necessarily invertible. The proof of this fact can be found, for example, in [2, Remark  2.6].

In order to simplify the notation we abbreviate [ ๐‘“ ๐œŽ ( ๐‘ก ) ] โˆ— and [ ๐‘“ โˆ— ( ๐‘ก ) ] ๐œŽ by ๐‘“ ๐œŽ โˆ— ( ๐‘ก ) . Similarly, instead of [ ๐‘“ ฮ” ( ๐‘ก ) ] โˆ— and [ ๐‘“ โˆ— ( ๐‘ก ) ] ฮ” we will use ๐‘“ ฮ” โˆ— ( ๐‘ก ) .

3. Time Scale Symplectic Systems

Let ๐’œ ( โ‹… ) , โ„ฌ ( โ‹… ) , ๐’ž ( โ‹… ) , ๐’Ÿ ( โ‹… ) , ๐’ฒ ( โ‹… ) be ๐‘› ร— ๐‘› piecewise rd-continuous functions on [ ๐‘Ž , โˆž ) ๐•‹ such that ๐’ฒ ( ๐‘ก ) โ‰ฅ 0 for all ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ ; that is, ๐’ฒ ( ๐‘ก ) is Hermitian and nonnegative definite, satisfying identity (1.8). In this paper we consider the linear system ( ๐’ฎ ๐œ† ) introduced in the previous section. This system can be written as ๐‘ง ฮ” ๎‚‹ ๐’ฒ ( ๐‘ก , ๐œ† ) = ๐’ฎ ( ๐‘ก ) ๐‘ง ( ๐‘ก , ๐œ† ) + ๐œ† ๐’ฅ ( ๐‘ก ) ๐‘ง ๐œŽ [ ) ( ๐‘ก , ๐œ† ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ๐•‹ , ( ๐’ฎ ๐œ† ) where the 2 ๐‘› ร— 2 ๐‘› matrix ๎‚‹ ๐’ฒ ( ๐‘ก ) is defined and has the property ๎‚‹ ๎‚ต ๎‚ถ ๎‚‹ ๎‚ต ๎‚ถ ๐’ฒ ( ๐‘ก ) โˆถ = ๐’ฒ ( ๐‘ก ) 0 0 0 , ๐’ฅ ๐’ฒ ( ๐‘ก ) = 0 0 โˆ’ ๐’ฒ ( ๐‘ก ) 0 . ( 3 . 1 ) The system ( ๐’ฎ ๐œ† ) can be written in the equivalent form ๐‘ง ฮ” ( [ ๐‘ก , ๐œ† ) = ๐’ฎ ( ๐‘ก , ๐œ† ) ๐‘ง ( ๐‘ก , ๐œ† ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 3 . 2 ) where the matrix ๐’ฎ ( ๐‘ก , ๐œ† ) is defined through the matrices ๐’ฎ ( ๐‘ก ) and ๎‚‹ ๐’ฒ ( ๐‘ก ) from (1.8) and (3.1) by ๐’ฎ ๎‚‹ ๐’ฒ [ ] = ๎‚ต [ ] ๎‚ถ . ( ๐‘ก , ๐œ† ) โˆถ = ๐’ฎ ( ๐‘ก ) + ๐œ† ๐’ฅ ( ๐‘ก ) ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก ) ๐’œ ( ๐‘ก ) โ„ฌ ( ๐‘ก ) ๐’ž ( ๐‘ก ) โˆ’ ๐œ† ๐’ฒ ( ๐‘ก ) ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’œ ( ๐‘ก ) ๐’Ÿ ( ๐‘ก ) โˆ’ ๐œ† ๐œ‡ ( ๐‘ก ) ๐’ฒ ( ๐‘ก ) โ„ฌ ( ๐‘ก ) ( 3 . 3 ) By using the identity in (1.8), a direct calculation shows that the matrix function ๐’ฎ ( โ‹… , โ‹… ) satisfies ๐’ฎ โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ + ๐’ฅ ๐’ฎ ๐‘ก , ๐œ† ๎‚ + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ๐’ฎ ๐‘ก , ๐œ† ๎‚ [ = 0 , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ๐œ† โˆˆ โ„‚ . ( 3 . 4 ) Here ๐’ฎ โˆ— ( ๐‘ก , ๐œ† ) = [ ๐’ฎ ( ๐‘ก , ๐œ† ) ] โˆ— , and ๐œ† is the usual conjugate number to ๐œ† .

Remark 3.1. The name time scale symplectic system or Hamiltonian system has been reserved in the literature for the system of the form ๐‘ง ฮ” ( [ ๐‘ก ) = ๐•Š ( ๐‘ก ) ๐‘ง ( ๐‘ก ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 3 . 5 ) in which the matrix function ๐•Š ( โ‹… ) satisfies the identity in (1.8); see [4447, 57], and compare also, for example, with [5861]. Since for a fixed ๐œ† , ๐œˆ โˆˆ โ„‚ the matrix ๐’ฎ ( ๐‘ก , ๐œ† ) from (3.3) satisfies ๐’ฎ โˆ— ( ๐‘ก , ๐œ† ) ๐’ฅ + ๐’ฅ ๐’ฎ ( ๐‘ก , ๐œˆ ) + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ๐’ฎ ( ๐‘ก , ๐œˆ ) = ๎‚ ๎€บ ๐œ† โˆ’ ๐œˆ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎€ป ๎‚‹ [ ] ( ๐‘ก ) ๐’ฒ ( ๐‘ก ) ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก ) , ( 3 . 6 ) it follows that the system ( ๐’ฎ ๐œ† ) is a true time scale symplectic system according to the above terminology only for ๐œ† โˆˆ โ„ , while strictly speaking ( ๐’ฎ ๐œ† ) is not a time scale symplectic system for ๐œ† โˆˆ โ„‚ โงต โ„ . However, since ( ๐’ฎ ๐œ† ) is a perturbation of the time scale symplectic system ( ๐’ฎ 0 ) and since the important properties of time scale symplectic systems needed in the presented Weyl-Titchmarsh theory, such as (3.4) or (3.8), are satisfied in an appropriate modification, we accept with the above understanding the same terminology for the system ( ๐’ฎ ๐œ† ) for any ๐œ† โˆˆ โ„‚ .

Equation (3.4) represents a fundamental identity for the theory of time scale symplectic systems ( ๐’ฎ ๐œ† ). Some important properties of the matrix ๐’ฎ ( ๐‘ก , ๐œ† ) are displayed below. Note that formula (3.7) is a generalization of [46, equation  (10.4)] to complex values of ๐œ† .

Lemma 3.2. Identity (3.4) is equivalent to the identity ๐’ฎ ๎‚€ ๐‘ก , ๐œ† ๎‚ ๐’ฅ + ๐’ฅ ๐’ฎ โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) + ๐œ‡ ( ๐‘ก ) ๐’ฎ ๐‘ก , ๐œ† ๎‚ ๐’ฅ ๐’ฎ โˆ— [ ( ๐‘ก , ๐œ† ) = 0 , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ๐œ† โˆˆ โ„‚ . ( 3 . 7 ) In this case for any ๐œ† โˆˆ โ„‚ we have ๎€บ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎€ป ๐’ฅ ๎‚ƒ ๎‚€ ( ๐‘ก , ๐œ† ) ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ๐‘ก , ๐œ† [ ๎‚ ๎‚„ = ๐’ฅ , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ ๎‚ƒ ๎‚€ , ( 3 . 8 ) ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ๐‘ก , ๐œ† ๐’ฅ ๎€บ ๎‚ ๎‚„ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎€ป [ ( ๐‘ก , ๐œ† ) = ๐’ฅ , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 3 . 9 ) and the matrices ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) and ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) are invertible with [ ] ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) โˆ’ 1 ๎‚ƒ = โˆ’ ๐’ฅ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎‚€ ๐‘ก , ๐œ† [ ๎‚ ๎‚„ ๐’ฅ , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ . ( 3 . 1 0 )

Proof. Let ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ and ๐œ† โˆˆ โ„‚ be fixed. If ๐‘ก is right-dense, that is, ๐œ‡ ( ๐‘ก ) = 0 , then identity (3.4) reduces to ๐’ฎ โˆ— ( ๐‘ก , ๐œ† ) ๐’ฅ + ๐’ฅ ๐’ฎ ( ๐‘ก , ๐œ† ) = 0 . Upon multiplying this equation by ๐’ฅ from the left and right side, we get identity (3.7) with ๐œ‡ ( ๐‘ก ) = 0 . If ๐‘ก is right scattered, that is, ๐œ‡ ( ๐‘ก ) > 0 , then (3.4) is equivalent to (3.8). It follows that the determinants of ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) and ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) are nonzero proving that these matrices are invertible with the inverse given by (3.10). Upon multiplying (3.8) by the invertible matrices [ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ] ๐’ฅ from the left and โˆ’ [ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ] โˆ’ 1 ๐’ฅ from the right and by using ๐’ฅ 2 = โˆ’ ๐ผ , we get formula (3.9), which is equivalent to (3.7) due to ๐œ‡ ( ๐‘ก ) > 0 .

Remark 3.3. Equation (3.10) allows writing the system ( ๐’ฎ ๐œ† ) in the equivalent adjoint form ๐‘ง ฮ” ( ๐‘ก , ๐œ† ) = ๐’ฅ ๐’ฎ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ ๐’ฅ ๐‘ง ๐œŽ [ ( ๐‘ก , ๐œ† ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ . ( 3 . 1 1 ) System (3.11) can be found, for example, in [47, Remark  3.1(iii)] or [50, equation  (3.2)] in the connection with optimality conditions for variational problems over time scales.

In the following result we show that (3.4) guarantees, among other properties, the existence and uniqueness of solutions of the initial value problems associated with ( ๐’ฎ ๐œ† ).

Theorem 3.4 (existence and uniqueness theorem). Let ๐œ† โˆˆ โ„‚ , ๐‘ก 0 โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ , and ๐‘ง 0 โˆˆ โ„‚ 2 ๐‘› be given. Then the initial value problem ( ๐’ฎ ๐œ† ) with ๐‘ง ( ๐‘ก 0 ) = ๐‘ง 0 has a unique solution ๐‘ง ( โ‹… , ๐œ† ) โˆˆ C 1 p r d on the interval [ ๐‘Ž , โˆž ) ๐•‹ .

Proof. The coefficient matrix of system ( ๐’ฎ ๐œ† ), or equivalently of system (3.2), is piecewise rd-continuous on [ ๐‘Ž , โˆž ) ๐•‹ . By Lemma 3.2, the matrix ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) is invertible for all ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ , which proves that the function ๐’ฎ ( โ‹… , ๐œ† ) is regressive on [ ๐‘Ž , โˆž ) ๐•‹ . Hence, the result follows from Remark 2.1.

If not specified otherwise, we use a common agreement that 2 ๐‘› -vector solutions of system ( ๐’ฎ ๐œ† ) and 2 ๐‘› ร— ๐‘› -matrix solutions of system ( ๐’ฎ ๐œ† ) are denoted by small letters and capital letters, respectively, typically by ๐‘ง ( โ‹… , ๐œ† ) or ฬƒ ๐‘ง ( โ‹… , ๐œ† ) and ๐‘ ( โ‹… , ๐œ† ) or ๎‚ ๐‘ ( โ‹… , ๐œ† ) .

Next we establish several identities involving solutions of system ( ๐’ฎ ๐œ† ) or solutions of two such systems with different spectral parameters. The first result is the Lagrange identity known in the special cases of continuous time linear Hamiltonian systems in [11, Theorem  4.1] or [8, equation  (2.23)], discrete linear Hamiltonian systems in [9, equation  (2.55)] or [14, Lemma  2.2], discrete symplectic systems in [1, Lemma  2.6] or [2, Lemma  2.3], and time scale linear Hamiltonian systems in [3, Lemma  3.5] and [5, Theorem  2.2].

Theorem 3.5 (Lagrange identity). Let ๐œ† , ๐œˆ โˆˆ โ„‚ and ๐‘š โˆˆ โ„• be given. If ๐‘ง ( โ‹… , ๐œ† ) and ๐‘ง ( โ‹… , ๐œˆ ) are 2 ๐‘› ร— ๐‘š solutions of systems ( ๐’ฎ ๐œ† ) and ( ๐’ฎ ๐œˆ ), respectively, then ๎€บ ๐‘ง โˆ— ๎€ป ( ๐‘ก , ๐œ† ) ๐’ฅ ๐‘ง ( ๐‘ก , ๐œˆ ) ฮ” = ๎‚€ ๎‚ ๐‘ง ๐œ† โˆ’ ๐œˆ ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† ) ๐’ฒ ( ๐‘ก ) ๐‘ง ๐œŽ [ ( ๐‘ก , ๐œˆ ) , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ . ( 3 . 1 2 )

Proof. Formula (3.12) follows from the time scales product rule (2.1) by using the relation ๐‘ง ๐œŽ ( ๐‘ก , ๐œ† ) = [ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ] ๐‘ง ( ๐‘ก , ๐œ† ) and identity (3.6).

As consequences of Theorem 3.5, we obtain the following.

Corollary 3.6. Let ๐œ† , ๐œˆ โˆˆ โ„‚ and ๐‘š โˆˆ โ„• be given. If ๐‘ง ( โ‹… , ๐œ† ) and ๐‘ง ( โ‹… , ๐œˆ ) are 2 ๐‘› ร— ๐‘š solutions of systems ( ๐’ฎ ๐œ† ) and ( ๐’ฎ ๐œˆ ), respectively, then for all ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ we have ๐‘ง โˆ— ( ๐‘ก , ๐œ† ) ๐’ฅ ๐‘ง ( ๐‘ก , ๐œˆ ) = ๐‘ง โˆ— ( ๎‚€ ๐‘Ž , ๐œ† ) ๐’ฅ ๐‘ง ( ๐‘Ž , ๐œˆ ) + ๎‚ ๎€œ ๐œ† โˆ’ ๐œˆ ๐‘ก ๐‘Ž ๐‘ง ๐œŽ โˆ— ( ๎‚‹ ๐‘  , ๐œ† ) ๐’ฒ ( ๐‘  ) ๐‘ง ๐œŽ ( ๐‘  , ๐œˆ ) ฮ” ๐‘  . ( 3 . 1 3 )

One can easily see that if ๐‘ง ( โ‹… , ๐œ† ) is a solution of system ( ๐’ฎ ๐œ† ), then ๐‘ง ( โ‹… , ๐œ† ) is a solution of system ( ๐’ฎ ๐œ† ). Therefore, Theorem 3.5 with ๐œˆ = ๐œ† yields a Wronskian-type property of solutions of system ( ๐’ฎ ๐œ† ).

Corollary 3.7. Let ๐œ† โˆˆ โ„‚ and ๐‘š โˆˆ โ„• be given. For any 2 ๐‘› ร— ๐‘š solution ๐‘ง ( โ‹… , ๐œ† ) of systems ( ๐’ฎ ๐œ† ) ๐‘ง โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ๐‘ง ๐‘ก , ๐œ† ๎‚ โ‰ก ๐‘ง โˆ— ๎‚€ ( ๐‘Ž , ๐œ† ) ๐’ฅ ๐‘ง ๐‘Ž , ๐œ† ๎‚ [ , i s c o n s t a n t o n ๐‘Ž , โˆž ) ๐•‹ . ( 3 . 1 4 )

The following result gives another interesting property of solutions of system ( ๐’ฎ ๐œ† ) and ( ๐’ฎ ๐œ† ).

Lemma 3.8. Let ๐œ† โˆˆ โ„‚ and ๐‘š โˆˆ โ„• be given. For any 2 ๐‘› ร— ๐‘š solutions ๐‘ง ( โ‹… , ๐œ† ) and ฬƒ ๐‘ง ( โ‹… , ๐œ† ) of system ( ๐’ฎ ๐œ† ), the 2 ๐‘› ร— 2 ๐‘› matrix function ๐พ ( โ‹… , ๐œ† ) defined by ๐พ ( ๐‘ก , ๐œ† ) โˆถ = ๐‘ง ( ๐‘ก , ๐œ† ) ฬƒ ๐‘ง โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ โˆ’ ฬƒ ๐‘ง ( ๐‘ก , ๐œ† ) ๐‘ง โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ [ , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 3 . 1 5 ) satisfies the dynamic equation ๐พ ฮ” [ ] ( ๐‘ก , ๐œ† ) = ๐’ฎ ( ๐‘ก , ๐œ† ) ๐พ ( ๐‘ก , ๐œ† ) + ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ๐พ ( ๐‘ก , ๐œ† ) ๐’ฎ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ [ , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ , ( 3 . 1 6 ) and the identities ๐พ โˆ— ( ๐‘ก , ๐œ† ) = โˆ’ ๐พ ( ๐‘ก , ๐œ† ) and ๐พ ๐œŽ [ ] ๎‚ƒ ( ๐‘ก , ๐œ† ) = ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ๐พ ( ๐‘ก , ๐œ† ) ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ โˆ— ๎‚€ ๐‘ก , ๐œ† [ ๎‚ ๎‚„ , ๐‘ก โˆˆ ๐‘Ž , โˆž ) ๐•‹ . ( 3 . 1 7 )

Proof. Having that ๐‘ง ( โ‹… , ๐œ† ) and ฬƒ ๐‘ง ( โ‹… , ๐œ† ) are solutions of system ( ๐’ฎ ๐œ† ), it follows that ๐‘ง ( โ‹… , ๐œ† ) and ฬƒ ๐‘ง ( โ‹… , ๐œ† ) are solutions of system ( ๐’ฎ ๐œ† ). The results then follow by direct calculations.

Remark 3.9. The content of Lemma 3.8 appears to be new both in the continuous and discrete time cases. Moreover, when the matrix function ๐พ ( โ‹… , ๐œ† ) โ‰ก ๐พ ( ๐œ† ) is constant, identity (3.17) yields for any right-scattered ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ that ๐’ฎ ( ๐‘ก , ๐œ† ) ๐พ ( ๐œ† ) + ๐พ ( ๐œ† ) ๐’ฎ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ๐พ ( ๐œ† ) ๐’ฎ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ = 0 . ( 3 . 1 8 ) It is interesting to note that this formula is very much like (3.7). More precisely, identity (3.7) is a consequence of (3.18) for the case of ๐พ ( ๐œ† ) โ‰ก ๐’ฅ .

Next we present properties of certain fundamental matrices ฮจ ( โ‹… , ๐œ† ) of system ( ๐’ฎ ๐œ† ), which are generalizations of the corresponding results in [46, Section  10.2] to complex ๐œ† . Some of these results can be proven under the weaker condition that the initial value of ฮจ ( ๐‘Ž , ๐œ† ) does depend on ๐œ† and satisfies ฮจ โˆ— ( ๐‘Ž , ๐œ† ) ๐’ฅ ฮจ ( ๐‘Ž , ๐œ† ) = ๐’ฅ . However, these more general results will not be needed in this paper.

Lemma 3.10. Let ๐œ† โˆˆ โ„‚ be fixed. If   ฮจ ( โ‹… , ๐œ† ) is a fundamental matrix of system ( ๐’ฎ ๐œ† ) such that ฮจ ( ๐‘Ž , ๐œ† ) is symplectic and independent of ๐œ† , then for any ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ we have ฮจ โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ฮจ ๐‘ก , ๐œ† ๎‚ = ๐’ฅ , ฮจ โˆ’ 1 ( ๐‘ก , ๐œ† ) = โˆ’ ๐’ฅ ฮจ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ ๐’ฅ , ฮจ ( ๐‘ก , ๐œ† ) ๐’ฅ ฮจ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ = ๐’ฅ . ( 3 . 1 9 )

Proof. Identity (3.19)(i) is a consequence of Corollary 3.7, in which we use the fact that ฮจ ( ๐‘Ž , ๐œ† ) is symplectic and independent of ๐œ† . The second identity in (3.19) follows from the first one, while the third identity is obtained from the equation ฮจ ( ๐‘ก , ๐œ† ) ฮจ โˆ’ 1 ( ๐‘ก , ๐œ† ) = ๐ผ .

Remark 3.11. If the fundamental matrix ๎‚ ฮจ ( โ‹… , ๐œ† ) = ( ๐‘ ( โ‹… , ๐œ† ) ๐‘ ( โ‹… , ๐œ† ) ) in Lemma 3.10 is partitioned into two 2 ๐‘› ร— ๐‘› blocks, then (3.19)(i) and (3.19)(iii) have, respectively, the form ๐‘ โˆ— ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ๐‘ ๐‘ก , ๐œ† ๎‚ = 0 , ๐‘ โˆ— ๎‚ ๐‘ ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ๐‘ก , ๐œ† ๎‚ ๎‚ ๐‘ = ๐ผ , โˆ— ๎‚ ๐‘ ๎‚€ ( ๐‘ก , ๐œ† ) ๐’ฅ ๐‘ก , ๐œ† ๎‚ ๎‚ ๐‘ = 0 , ( 3 . 2 0 ) ๐‘ ( ๐‘ก , ๐œ† ) โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ โˆ’ ๎‚ ๐‘ ( ๐‘ก , ๐œ† ) ๐‘ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ = ๐’ฅ . ( 3 . 2 1 ) Observe that the matrix on the left-hand side of (3.21) represents a constant matrix ๐พ ( ๐‘ก , ๐œ† ) from Lemma 3.8 and Remark 3.9.

Corollary 3.12. Under the conditions of Lemma 3.10, for any ๐‘ก โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ , we have ฮจ ๐œŽ ( ๐‘ก , ๐œ† ) ๐’ฅ ฮจ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ = [ ] ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ๐’ฅ , ( 3 . 2 2 ) which in the notation of Remark 3.11 has the form ๐‘ ๐œŽ ๎‚ ๐‘ ( ๐‘ก , ๐œ† ) โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ โˆ’ ๎‚ ๐‘ ๐œŽ ( ๐‘ก , ๐œ† ) ๐‘ โˆ— ๎‚€ ๐‘ก , ๐œ† ๎‚ = [ ] ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ๐’ฅ . ( 3 . 2 3 )

Proof. Identity (3.22) follows from the equation ฮจ ๐œŽ ( ๐‘ก , ๐œ† ) = [ ๐ผ + ๐œ‡ ( ๐‘ก ) ๐’ฎ ( ๐‘ก , ๐œ† ) ] ฮจ ( ๐‘ก , ๐œ† ) by applying formula (3.19)(ii).

4. ๐‘€ ( ๐œ† ) -Function for Regular Spectral Problem

In this section we consider the regular spectral problem on the time scale interval [ ๐‘Ž , ๐‘ ] ๐•‹ with some fixed ๐‘ โˆˆ ( ๐‘Ž , โˆž ) ๐•‹ . We will specify the corresponding boundary conditions in terms of complex ๐‘› ร— 2 ๐‘› matrices from the set ๎€ฝ ฮ“ โˆถ = ๐›ผ โˆˆ โ„‚ ๐‘› ร— 2 ๐‘› , ๐›ผ ๐›ผ โˆ— = ๐ผ , ๐›ผ ๐’ฅ ๐›ผ โˆ— ๎€พ = 0 . ( 4 . 1 ) The two defining conditions for ๐›ผ โˆˆ โ„‚ ๐‘› ร— 2 ๐‘› in (4.1) imply that the 2 ๐‘› ร— 2 ๐‘› matrix ( ๐›ผ โˆ— โˆ’ ๐’ฅ ๐›ผ โˆ— ) is unitary and symplectic. This yields the identity ๎€ท ๐›ผ โˆ— โˆ’ ๐’ฅ ๐›ผ โˆ— ๎€ธ ๎‚ต ๐›ผ ๎‚ถ ๐›ผ ๐’ฅ = ๐ผ โˆˆ โ„‚ 2 ๐‘› ร— 2 ๐‘› , t h a t i s , ๐›ผ โˆ— ๐›ผ โˆ’ ๐’ฅ ๐›ผ โˆ— ๐›ผ ๐’ฅ = ๐ผ . ( 4 . 2 ) The last equation also implies, compare with [60, Remark  2.1.2], that K e r ๐›ผ = I m ๐’ฅ ๐›ผ โˆ— . ( 4 . 3 )

Let ๐›ผ , ๐›ฝ โˆˆ ฮ“ be fixed and consider the boundary value problem ๎€ท ๐’ฎ ๐œ† ๎€ธ , ๐›ผ ๐‘ง ( ๐‘Ž , ๐œ† ) = 0 , ๐›ฝ ๐‘ง ( ๐‘ , ๐œ† ) = 0 . ( 4 . 4 ) Our first result shows that the boundary conditions in (4.4) are equivalent with the boundary conditions phrased in terms of the images of the 2 ๐‘› ร— 2 ๐‘› matrices ๐‘… ๐‘Ž ๎€ท โˆถ = ๐’ฅ ๐›ผ โˆ— 0 ๎€ธ , ๐‘… ๐‘ ๎€ท โˆถ = 0 โˆ’ ๐’ฅ ๐›ฝ โˆ— ๎€ธ , ( 4 . 5 ) which satisfy ๐‘… โˆ— ๐‘Ž ๐’ฅ ๐‘… ๐‘Ž = 0 , ๐‘… โˆ— ๐‘ ๐’ฅ ๐‘… ๐‘ = 0 , and r a n k ( ๐‘… โˆ— ๐‘Ž ๐‘… โˆ— ๐‘ ) = 2 ๐‘› .

Lemma 4.1. Let ๐›ผ , ๐›ฝ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ be fixed. A solution ๐‘ง ( โ‹… , ๐œ† ) of system ( ๐’ฎ ๐œ† ) satisfies the boundary conditions in (4.4) if and only if there exists a unique vector ๐œ‰ โˆˆ โ„‚ 2 ๐‘› such that ๐‘ง ( ๐‘Ž , ๐œ† ) = ๐‘… ๐‘Ž ๐œ‰ , ๐‘ง ( ๐‘ , ๐œ† ) = ๐‘… ๐‘ ๐œ‰ . ( 4 . 6 )

Proof. Assume that (4.4) holds. Identity (4.3) implies the existence of vectors ๐œ‰ ๐‘Ž , ๐œ‰ ๐‘ โˆˆ โ„‚ ๐‘› such that ๐‘ง ( ๐‘Ž , ๐œ† ) = โˆ’ ๐’ฅ ๐›ผ โˆ— ๐œ‰ ๐‘Ž and ๐‘ง ( ๐‘ , ๐œ† ) = โˆ’ ๐’ฅ ๐›ฝ โˆ— ๐œ‰ ๐‘ . It follows that ๐‘ง ( โ‹… , ๐œ† ) satisfies (4.6) with ๐œ‰ โˆถ = ( โˆ’ ๐œ‰ โˆ— ๐‘Ž ๐œ‰ โˆ— ๐‘ ) โˆ— . It remains to prove that ๐œ‰ is unique such a vector. If ๐‘ง ( โ‹… , ๐œ† ) satisfies (4.6) and also ๐‘ง ( ๐‘Ž , ๐œ† ) = ๐‘… ๐‘Ž ๐œ and ๐‘ง ( ๐‘ , ๐œ† ) = ๐‘… ๐‘ ๐œ for some ๐œ‰ , ๐œ โˆˆ โ„‚ 2 ๐‘› , then ๐‘… ๐‘Ž ( ๐œ‰ โˆ’ ๐œ ) = 0 and ๐‘… ๐‘ ( ๐œ‰ โˆ’ ๐œ ) = 0 . Hence, ๐’ฅ ๐›ผ โˆ— ( ๐ผ 0 ) ( ๐œ‰ โˆ’ ๐œ ) = 0 and โˆ’ ๐’ฅ ๐›ฝ โˆ— ( 0 ๐ผ ) ( ๐œ‰ โˆ’ ๐œ ) = 0 . If we multiply the latter two equalities by ๐›ผ ๐’ฅ and ๐›ฝ ๐’ฅ , respectively, and use ๐›ผ ๐›ผ โˆ— = ๐ผ = ๐›ฝ ๐›ฝ โˆ— , then we obtain ( ๐ผ 0 ) ( ๐œ‰ โˆ’ ๐œ ) = 0 and ( 0 ๐ผ ) ( ๐œ‰ โˆ’ ๐œ ) = 0 . This yields ๐œ‰ โˆ’ ๐œ = 0 , which shows that the vector ๐œ‰ in (4.6) is unique. The opposite direction, that is, that (4.6) implies (4.4), is trivial.

Following the standard terminology, see, for example, [62, 63], a number ๐œ† โˆˆ โ„‚ is an eigenvalue of (4.4) if this boundary value problem has a solution ๐‘ง ( โ‹… , ๐œ† ) โ‰ข 0 . In this case the function ๐‘ง ( โ‹… , ๐œ† ) is called the eigenfunction corresponding to the eigenvalue ๐œ† , and the dimension of the space of all eigenfunctions corresponding to ๐œ† (together with the zero function) is called the geometric multiplicity of ๐œ† .

Given ๐›ผ โˆˆ ฮ“ , we will utilize from now on the fundamental matrix ฮจ ( โ‹… , ๐œ† , ๐›ผ ) of system ( ๐’ฎ ๐œ† ) satisfying the initial condition from (4.4), that is, ฮจ ฮ” [ ] ( ๐‘ก , ๐œ† , ๐›ผ ) = ๐’ฎ ( ๐‘ก , ๐œ† ) ฮจ ( ๐‘ก , ๐œ† , ๐›ผ ) , ๐‘ก โˆˆ ๐‘Ž , ๐œŒ ( ๐‘ ) ๐•‹ ๎€ท ๐›ผ , ฮจ ( ๐‘Ž , ๐œ† , ๐›ผ ) = โˆ— โˆ’ ๐’ฅ ๐›ผ โˆ— ๎€ธ . ( 4 . 7 ) Then ฮจ ( ๐‘Ž , ๐œ† , ๐›ผ ) does not depend on ๐œ† , and it is symplectic and unitary with the inverse ฮจ โˆ’ 1 ( ๐‘Ž , ๐œ† , ๐›ผ ) = ฮจ โˆ— ( ๐‘Ž , ๐œ† , ๐›ผ ) . Hence, the properties of fundamental matrices derived earlier in Lemma 3.10, Remark 3.11, and Corollary 3.12 apply for the matrix function ฮจ ( โ‹… , ๐œ† , ๐›ผ ) .

The following assumption will be imposed in this section when studying the regular spectral problem.

Hypothesis 4.2. For every ๐œ† โˆˆ โ„‚ , we have ๎€œ ๐‘ ๐‘Ž ฮจ ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† , ๐›ผ ) ๐’ฒ ( ๐‘ก ) ฮจ ๐œŽ ( ๐‘ก , ๐œ† , ๐›ผ ) ฮ” ๐‘ก > 0 . ( 4 . 8 )

Condition (4.8) can be written in the equivalent form as ๎€œ ๐‘ ๐‘Ž ๐‘ง ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† ) ๐’ฒ ( ๐‘ก ) ๐‘ง ๐œŽ ( ๐‘ก , ๐œ† ) ฮ” ๐‘ก > 0 , ( 4 . 9 ) for every nontrivial solution ๐‘ง ( โ‹… , ๐œ† ) of system ( ๐’ฎ ๐œ† ). Assumptions (4.8) and (4.9) are equivalent by a simple argument using the uniqueness of solutions of system ( ๐’ฎ ๐œ† ). The latter form (4.9) has been widely used in the literature, such as in the continuous time case in [8, Hypothesis  2.2], [30, equation  (1.3)], [26, equation  (2.3)], in the discrete time case in [9, Condition  (2.16)], [14, equation  (1.7)], [1, Assumption  2.2], [2, Hypothesis  2.4], and in the time scale Hamiltonian case in [3, Assumption  3] and [5, Condition  (3.9)].

Following Remark 3.11, we partition the fundamental matrix ฮจ ( โ‹… , ๐œ† , ๐›ผ ) as ๎€ท ๐‘ ๎‚ ๐‘ ๎€ธ ฮจ ( โ‹… , ๐œ† , ๐›ผ ) = ( โ‹… , ๐œ† , ๐›ผ ) ( โ‹… , ๐œ† , ๐›ผ ) , ( 4 . 1 0 ) where ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) and ๎‚ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) are the 2 ๐‘› ร— ๐‘› solutions of system ( ๐’ฎ ๐œ† ) satisfying ๐‘ ( ๐‘Ž , ๐œ† , ๐›ผ ) = ๐›ผ โˆ— and ๎‚ ๐‘ ( ๐‘Ž , ๐œ† , ๐›ผ ) = โˆ’ ๐’ฅ ๐›ผ โˆ— . With the notation ฮ› ( ๐œ† , ๐›ผ , ๐›ฝ ) โˆถ = ฮจ ( ๐‘ , ๐œ† , ๐›ผ ) ฮจ โˆ— ( ๐‘Ž , ๐œ† , ๐›ผ ) ๐‘… ๐‘Ž โˆ’ ๐‘… ๐‘ = ๎€ท โˆ’ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ฅ ๐›ฝ โˆ— ๎€ธ , ( 4 . 1 1 ) we have the classical characterization of the eigenvalues of (4.4); see, for example, the continuous time in [64, Chapter  4], the discrete time in [14, Theorem  2.3, Lemma  2.4], [2, Lemma  2.9, Theorem  2.11], and the time scale case in [62, Lemma  3], [63, Corollary  1].

Proposition 4.3. For ๐›ผ , ๐›ฝ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ , we have with notation (4.11) the following. (i)The number ๐œ† is an eigenvalue of (4.4) if and only if d e t ฮ› ( ๐œ† , ๐›ผ , ๐›ฝ ) = 0 . (ii)The algebraic multiplicity of the eigenvalue ๐œ† , that is, the number d e f ฮ› ( ๐œ† , ๐›ผ , ๐›ฝ ) , is equal to the geometric multiplicity of ๐œ† . (iii)Under Hypothesis 4.2, the eigenvalues of (4.4) are real, and the eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the semi-inner product โŸจ ๐‘ง ( โ‹… , ๐œ† ) , ๐‘ง ( โ‹… , ๐œˆ ) โŸฉ ๐’ฒ , ๐‘ ๎€œ โˆถ = ๐‘ ๐‘Ž ๐‘ง ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† ) ๐’ฒ ( ๐‘ก ) ๐‘ง ๐œŽ ( ๐‘ก , ๐œˆ ) ฮ” ๐‘ก . ( 4 . 1 2 )

Proof. The arguments are here standard, and we refer to [44, Section  5], [63, Corollary  1], [3, Theorem  3.6].

The next algebraic characterization of the eigenvalues of (4.4) is more appropriate for the development of the Weyl-Titchmarsh theory for (4.4), since it uses the matrix ๐›ฝ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) which has dimension ๐‘› instead of using the matrix ฮ› ( ๐œ† , ๐›ผ , ๐›ฝ ) which has dimension 2 ๐‘› . Results of this type can be found in special cases of system ( ๐’ฎ ๐œ† ) in [8, Lemma  2.5], [11, Theorem  4.1], [9, Lemma  2.8], [14, Lemma  3.1], [1, Lemma  2.5], [3, Theorem  3.4], and [2, Lemma  3.1].

Lemma 4.4. Let ๐›ผ , ๐›ฝ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ be fixed. Then ๐œ† is an eigenvalue of (4.4) if and only if ๎‚ d e t ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) = 0 . In this case the algebraic and geometric multiplicities of ๐œ† are equal to ๎‚ d e f ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) .

Proof. One can follow the same arguments as in the proof of the corresponding discrete symplectic case in [2, Lemma  3.1]. However, having the result of Proposition 4.3, we can proceed directly by the methods of linear algebra. In this proof we abbreviate ฮ› โˆถ = ฮ› ( ๐œ† , ๐›ผ , ๐›ฝ ) and ๎‚ ๎‚ ๐‘ โˆถ = ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) . Assume that ฮ› is singular, that is, โˆ’ ๎‚ ๐‘ ๐‘ + ๐’ฅ ๐›ฝ โˆ— ๐‘‘ = 0 for some vectors ๐‘ , ๐‘‘ โˆˆ โ„‚ ๐‘› , not both zero. Then ๎‚ ๐‘ ๐‘ = ๐’ฅ ๐›ฝ โˆ— ๐‘‘ , which yields that ๐›ฝ ๎‚ ๐‘ ๐‘ = 0 . If ๐‘ = 0 , then ๐’ฅ ๐›ฝ โˆ— ๐‘‘ = 0 , which implies upon the multiplication by ๐›ฝ ๐’ฅ from the left that ๐‘‘ = 0 . Since not both ๐‘ and ๐‘‘ can be zero, it follows that ๐‘ โ‰  0 and the matrix ๐›ฝ ๎‚ ๐‘ is singular. Conversely, if ๐›ฝ ๎‚ ๐‘ ๐‘ = 0 for some nonzero vector ๐‘ โˆˆ โ„‚ ๐‘› , then โˆ’ ๎‚ ๐‘ ๐‘ + ๐’ฅ ๐›ฝ โˆ— ๐‘‘ = 0 ; that is, ฮ› is singular, with the vector ๎‚ ๐‘‘ โˆถ = โˆ’ ๐›ฝ ๐’ฅ ๐‘ ๐‘ . Indeed, by using identity (4.2) we have ๐’ฅ ๐›ฝ โˆ— ๐‘‘ = โˆ’ ๐’ฅ ๐›ฝ โˆ— ๎‚ ๐›ฝ ๐’ฅ ๐‘ ๐‘ = ( ๐ผ โˆ’ ๐›ฝ โˆ— ๎‚ ๎‚ ๐›ฝ ) ๐‘ ๐‘ = ๐‘ ๐‘ . From the above we can also see that the number of linearly independent vectors in ๎‚ ๐‘ K e r ๐›ฝ is the same as the number of linearly independent vectors in K e r ฮ› . Therefore, by Proposition 4.3(ii), the algebraic and geometric multiplicities of ๐œ† as an eigenvalue of (4.4) are equal to ๎‚ ๐‘ d e f ๐›ฝ .

Since the eigenvalues of (4.4) are real, it follows that the matrix ๐›ฝ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible for every ๐œ† โˆˆ โ„‚ except for at most ๐‘› real numbers. This motivates the definition of the ๐‘€ ( ๐œ† ) -function for the regular spectral problem.

Definition 4.5 ( ๐‘€ ( ๐œ† ) -function). Let ๐›ผ , ๐›ฝ โˆˆ ฮ“ . Whenever the matrix ๐›ฝ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible for some value ๐œ† โˆˆ โ„‚ , we define the Weyl-Titchmarsh ๐‘€ ( ๐œ† ) -function as the ๐‘› ร— ๐‘› matrix ๎‚ƒ ๐›ฝ ๎‚ ๎‚„ ๐‘€ ( ๐œ† ) = ๐‘€ ( ๐œ† , ๐‘ ) = ๐‘€ ( ๐œ† , ๐‘ , ๐›ผ , ๐›ฝ ) โˆถ = โˆ’ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) โˆ’ 1 ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) . ( 4 . 1 3 )

The above definition of the ๐‘€ ( ๐œ† ) -function is a generalization of the corresponding definitions for the continuous and discrete linear Hamiltonian and symplectic systems in [8, Definition  2.6], [9, Definition  2.9], [14, equation  (3.10)], [1, page  2859], [2, Definition  3.2] and time scale linear Hamiltonian systems in [3, equation  (4.1)]. The dependence of the ๐‘€ ( ๐œ† ) -function on ๐‘ , ๐›ผ , and ๐›ฝ will be suppressed in the notation, and ๐‘€ ( ๐œ† , ๐‘ ) or ๐‘€ ( ๐œ† , ๐‘ , ๐›ผ , ๐›ฝ ) will be used only in few situations when we emphasize the dependence on ๐‘ (such as at the end of Section 5) or on ๐›ผ and ๐›ฝ (as in Lemma 4.14). By [65, Corollary  4.5], see also [44, Remark  2.2], the ๐‘€ ( โ‹… ) -function is an entire function in ๐œ† . Another important property of the ๐‘€ ( ๐œ† ) -function is established in the following.

Lemma 4.6. Let ๐›ผ , ๐›ฝ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . Then ๐‘€ โˆ— ๎‚€ ( ๐œ† ) = ๐‘€ ๐œ† ๎‚ . ( 4 . 1 4 )

Proof. We abbreviate ๐‘ ( ๐œ† ) โˆถ = ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) and ๎‚ ๎‚ ๐‘ ( ๐œ† ) โˆถ = ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) . By using the definition of ๐‘€ ( ๐œ† ) in (4.13) and identity (3.21), we have ๐‘€ โˆ— ๎‚€ ( ๐œ† ) โˆ’ ๐‘€ ๐œ† ๎‚ = ๎‚ƒ ๐›ฝ ๎‚ ๐‘ ๎‚€ ๐œ† ๎‚ ๎‚„ โˆ’ 1 ๐›ฝ ๎‚ƒ ๐‘ ๎‚€ ๐œ† ๎‚ ๎‚ ๐‘ โˆ— ๎‚ ๐‘ ๎‚€ ( ๐œ† ) โˆ’ ๐œ† ๎‚ ๐‘ โˆ— ๎‚„ ๐›ฝ ( ๐œ† ) โˆ— ๎‚ƒ ๐›ฝ ๎‚ ๎‚„ ๐‘ ( ๐œ† ) โˆ— โˆ’ 1 ( 3 . 2 1 ) = ๎‚ƒ ๐›ฝ ๎‚ ๐‘ ๎‚€ ๐œ† ๎‚ ๎‚„ โˆ’ 1 ๐›ฝ ๐’ฅ ๐›ฝ โˆ— ๎‚ƒ ๐›ฝ ๎‚ ๎‚„ ๐‘ ( ๐œ† ) โˆ— โˆ’ 1 = 0 , ( 4 . 1 5 ) because ๐›ฝ โˆˆ ฮ“ . Hence, equality (4.14) holds true.

The following solution plays an important role in particular in the results concerning the square integrable solutions of system ( ๐’ฎ ๐œ† ).

Definition 4.7 (Weyl solution). For any matrix ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› , we define the so-called Weyl solution of system ( ๐’ฎ ๐œ† ) by ๐’ณ ๎€ท ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ) โˆถ = ฮจ ( โ‹… , ๐œ† , ๐›ผ ) ๐ผ ๐‘€ โˆ— ๎€ธ โˆ— ๎‚ ๐‘ = ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) + ( โ‹… , ๐œ† , ๐›ผ ) ๐‘€ , ( 4 . 1 6 ) where ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) and ๎‚ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) are defined in (4.10).

The function ๐’ณ ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ) , being a linear combination of two solutions of system ( ๐’ฎ ๐œ† ), is also a solution of this system. Moreover, ๐›ผ ๐’ณ ( ๐‘Ž , ๐œ† , ๐›ผ , ๐‘€ ) = ๐ผ , and, if ๐›ฝ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible, then ๐›ฝ ๎‚ ๎‚ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) [ ๐‘€ โˆ’ ๐‘€ ( ๐œ† ) ] . Consequently, if we take ๐‘€ โˆถ = ๐‘€ ( ๐œ† ) in Definition 4.7, then ๐›ฝ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ( ๐œ† ) ) = 0 ; that is, the Weyl solution ๐’ณ ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ( ๐œ† ) ) satisfies the right endpoint boundary condition in (4.4).

Following the corresponding notions in [8, equation  (2.18)], [9, equation  (2.51)], [14, page  471], [1, page  2859], [2, equation  (3.13)], [3, equation  (4.2)], we define the Hermitian ๐‘› ร— ๐‘› matrix function โ„ฐ ( ๐‘€ ) for system ( ๐’ฎ ๐œ† ).

Definition 4.8. For a fixed ๐›ผ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ , we define the matrix function โ„ฐ โˆถ โ„‚ ๐‘› ร— ๐‘› โŸถ โ„‚ ๐‘› ร— ๐‘› , โ„ฐ ( ๐‘€ ) = โ„ฐ ( ๐‘€ , ๐‘ ) โˆถ = ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฅ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) , ( 4 . 1 7 ) where ๐›ฟ ( ๐œ† ) โˆถ = s g n I m ( ๐œ† ) .

For brevity we suppress the dependence of the function โ„ฐ ( โ‹… ) on ๐‘ and ๐œ† . In few cases we will need โ„ฐ ( ๐‘€ ) depending on ๐‘ (as in Theorem 5.1 and Definition 6.2) and in such situations we will use the notation โ„ฐ ( ๐‘€ , ๐‘ ) . Since ( ๐‘– ๐’ฅ ) โˆ— = ๐‘– ๐’ฅ , it follows that โ„ฐ ( ๐‘€ ) is a Hermitian matrix for any ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› . Moreover, from Corollary 3.6, we obtain the identity | | | | ๎€œ โ„ฐ ( ๐‘€ ) = โˆ’ 2 ๐›ฟ ( ๐œ† ) I m ( ๐‘€ ) + 2 I m ( ๐œ† ) ๐‘ ๐‘Ž ๐’ณ ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฒ ( ๐‘ก ) ๐’ณ ๐œŽ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ฮ” ๐‘ก , ( 4 . 1 8 ) where we used the fact that ๐’ณ โˆ— ( ๐‘Ž , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฅ ๐’ณ ( ๐‘Ž , ๐œ† , ๐›ผ , ๐‘€ ) ( 4 . 7 ) = ๐‘€ โˆ’ ๐‘€ โˆ— = 2 ๐‘– I m ( ๐‘€ ) . ( 4 . 1 9 )

Next we define the Weyl disk and Weyl circle for the regular spectral problem. The geometric characterizations of the Weyl disk and Weyl circle in terms of the contractive or unitary matrices which justify the terminology “disk” or “circle” will be presented in Section 5.

Definition 4.9 (Weyl disk and Weyl circle). For a fixed ๐›ผ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ , the set ๐ท ๎€ฝ ( ๐œ† ) = ๐ท ( ๐œ† , ๐‘ ) โˆถ = ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› ๎€พ , โ„ฐ ( ๐‘€ ) โ‰ค 0 , ( 4 . 2 0 ) is called the Weyl disk, and the set ๐ถ ๎€ฝ ( ๐œ† ) = ๐ถ ( ๐œ† , ๐‘ ) โˆถ = ๐œ• ๐ท ( ๐œ† ) = ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› ๎€พ , โ„ฐ ( ๐‘€ ) = 0 , ( 4 . 2 1 ) is called the Weyl circle.

The dependence of the Weyl disk and Weyl circle on ๐‘ will be again suppressed. In the following result we show that the Weyl circle consists of precisely those matrices ๐‘€ ( ๐œ† ) with ๐›ฝ โˆˆ ฮ“ . This result generalizes the corresponding statements in [8, Lemma  2.8], [9, Lemma  2.13], [14, Lemma  3.3], [1, Theorem  3.1], [2, Theorem  3.6], and [3, Theorem  4.2].

Theorem 4.10. Let ๐›ผ โˆˆ ฮ“ , ๐œ† โˆˆ โ„‚ โงต โ„ , and ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› . The matrix ๐‘€ belongs to the Weyl circle ๐ถ ( ๐œ† ) if and only if there exists ๐›ฝ โˆˆ ฮ“ such that ๐›ฝ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = 0 . In this case and under Hypothesis 4.2, we have with such a matrix ๐›ฝ that ๐‘€ = ๐‘€ ( ๐œ† ) as defined in (4.13).

Proof. Assume that ๐‘€ โˆˆ ๐ถ ( ๐œ† ) , that is, โ„ฐ ( ๐‘€ ) = 0 . Then, with the vector ๐›ฝ โˆถ = ๐’ณ โˆ— ๎€ท ( ๐‘ ) ๐’ฅ = ๐ผ ๐‘€ โˆ— ๎€ธ ฮจ โˆ— ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ฅ โˆˆ โ„‚ ๐‘› ร— 2 ๐‘› , ( 4 . 2 2 ) where ๐’ณ ( ๐‘ ) denotes ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) , we have ๐›ฝ ๐’ณ ( ๐‘ ) = ๐’ณ โˆ— ๎‚ธ 1 ( ๐‘ ) ๐’ฅ ๐’ณ ( ๐‘ ) = ๎‚น ( ๐‘– ๐›ฟ ( ๐œ† ) ) โ„ฐ ( ๐‘€ ) = 0 . ( 4 . 2 3 ) Moreover, r a n k ๐›ฝ = ๐‘› , because the matrices ฮจ ( ๐‘ , ๐œ† , ๐›ผ ) and ๐’ฅ are invertible and r a n k ( ๐ผ ๐‘€ โˆ— ) = ๐‘› . In addition, the identity ๐’ฅ โˆ— = ๐’ฅ โˆ’ 1 yields ๐›ฝ ๐’ฅ ๐›ฝ โˆ— = ๐’ณ โˆ— ( ๐‘ ) ๐’ฅ ๐’ณ ( ๐‘ ) ( 4 . 2 3 ) = 0 . ( 4 . 2 4 ) Now, if the condition ๐›ฝ ๐›ฝ โˆ— = ๐ผ is not satisfied, then we replace ๐›ฝ by ฬƒ ๐›ฝ โˆถ = ( ๐›ฝ ๐›ฝ โˆ— ) โˆ’ 1 / 2 ๐›ฝ (note that ๐›ฝ ๐›ฝ โˆ— > 0 , so that ( ๐›ฝ ๐›ฝ โˆ— ) โˆ’ 1 / 2 is well defined), and in this case ฬƒ ๎€ท ๐›ฝ ๐’ณ ( ๐‘ ) = ๐›ฝ ๐›ฝ โˆ— ๎€ธ โˆ’ 1 / 2 ๐›ฝ ๐’ณ ( ๐‘ ) ( 4 . 2 3 ) ฬƒ ฬƒ ๐›ฝ = 0 , ๐›ฝ ๐’ฅ โˆ— = ๎€ท ๐›ฝ ๐›ฝ โˆ— ๎€ธ โˆ’ 1 / 2 ๐›ฝ ๐’ฅ ๐›ฝ โˆ— ๎€ท ๐›ฝ ๐›ฝ โˆ— ๎€ธ โˆ’ 1 / 2 ( 4 . 2 4 ) ฬƒ ๐›ฝ ฬƒ ๐›ฝ = 0 , โˆ— = ๎€ท ๐›ฝ ๐›ฝ โˆ— ๎€ธ โˆ’ 1 / 2 ๐›ฝ ๐›ฝ โˆ— ๎€ท ๐›ฝ ๐›ฝ โˆ— ๎€ธ โˆ’ 1 / 2 = ๐ผ . ( 4 . 2 5 ) Conversely, suppose that for a given ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› there exists ๐›ฝ โˆˆ ฮ“ such that ๐›ฝ ๐’ณ ( ๐‘ ) = 0 . Then from (4.3) it follows that ๐’ณ ( ๐‘ ) = ๐’ฅ ๐›ฝ โˆ— ๐‘ƒ for the matrix ๐‘ƒ โˆถ = โˆ’ ๐›ฝ ๐’ฅ ๐’ณ ( ๐‘ ) โˆˆ โ„‚ ๐‘› ร— ๐‘› . Hence, โ„ฐ ( ๐‘€ ) = ๐‘– ๐›ฟ ( ๐œ† ) ๐‘ƒ โˆ— ๐›ฝ ๐’ฅ โˆ— ๐’ฅ ๐’ฅ ๐›ฝ โˆ— ๐‘ƒ = ๐‘– ๐›ฟ ( ๐œ† ) ๐‘ƒ โˆ— ๐›ฝ ๐’ฅ ๐›ฝ โˆ— ๐‘ƒ = 0 , ( 4 . 2 6 ) that is, ๐‘€ โˆˆ ๐ถ ( ๐œ† ) . Finally, since ๐œ† โˆˆ โ„‚ โงต โ„ , then by Proposition 4.3(iii) the number ๐œ† is not an eigenvalue of (4.4), which by Lemma 4.4 shows that the matrix ๐›ฝ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible. The definition of the Weyl solution in (4.16) then yields ๎‚ ๐‘ ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) + ๐›ฝ ( ๐‘ , ๐œ† , ๐›ผ ) ๐‘€ = ๐›ฝ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = 0 , ( 4 . 2 7 ) which implies that ๎‚ ๐‘€ = โˆ’ [ ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) ] โˆ’ 1 ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) = ๐‘€ ( ๐œ† ) .

Remark 4.11. The matrix ๐‘ƒ โˆถ = โˆ’ ๐›ฝ ๐’ฅ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) โˆˆ โ„‚ ๐‘› ร— ๐‘› from the proof of Theorem 4.10 is invertible. This fact was not needed in that proof. However, we show that ๐‘ƒ is invertible because this argument will be used in the proof of Lemma 4.14. First we prove that K e r ๐‘ƒ = K e r ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) . For if ๐‘ƒ ๐‘‘ = 0 for some ๐‘‘ โˆˆ โ„‚ ๐‘› , then from identity (4.2) we get ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) ๐‘‘ = ( ๐ผ โˆ’ ๐›ฝ โˆ— ๐›ฝ ) ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) ๐‘‘ = ๐’ฅ ๐›ฝ โˆ— ๐‘ƒ ๐‘‘ = 0 . Therefore, K e r ๐‘ƒ โŠ† K e r ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) . The opposite inclusion follows by the definition of ๐‘ƒ . And since, by (4.16), r a n k ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = r a n k ( ๐ผ ๐‘€ โˆ— ) โˆ— = ๐‘› , it follows that K e r ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = { 0 } . Hence, K e r ๐‘ƒ = { 0 } as well; that is, the matrix ๐‘ƒ is invertible.

The next result contains a characterization of the matrices ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› which lie “inside” the Weyl disk ๐ท ( ๐œ† ) . In the previous result (Theorem 4.10) we have characterized the elements of the boundary of the Weyl disk ๐ท ( ๐œ† ) , that is, the elements of the Weyl circle ๐ถ ( ๐œ† ) , in terms of the matrices ๐›ฝ โˆˆ ฮ“ . For such ๐›ฝ we have ๐›ฝ ๐’ฅ ๐›ฝ โˆ— = 0 , which yields ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— = 0 . Comparing with that statement we now utilize the matrices ๐›ฝ โˆˆ โ„‚ ๐‘› ร— 2 ๐‘› which satisfy ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— > 0 . In the special cases of the continuous and discrete time, this result can be found in [8, Lemma  2.13], [9, Lemma  2.18], and [2, Theorem  3.13].

Theorem 4.12. Let ๐›ผ โˆˆ ฮ“ , ๐œ† โˆˆ โ„‚ โงต โ„ , and ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› . The matrix ๐‘€ satisfies โ„ฐ ( ๐‘€ ) < 0 if and only if there exists ๐›ฝ โˆˆ โ„‚ ๐‘› ร— 2 ๐‘› such that ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— > 0 and ๐›ฝ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = 0 . In this case and under Hypothesis 4.2, we have with such a matrix ๐›ฝ that ๐‘€ = ๐‘€ ( ๐œ† ) as defined in (4.13) and ๐›ฝ may be chosen so that ๐›ฝ ๐›ฝ โˆ— = ๐ผ .

Proof. For ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› consider on [ ๐‘Ž , ๐‘ ] ๐•‹ the Weyl solution ๎‚ต ๐’ณ ๐’ณ ( โ‹… ) โˆถ = ๐’ณ ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ) = 1 ๐’ณ ( โ‹… ) 2 ๎‚ถ ( โ‹… ) , w i t h ๐‘› ร— ๐‘› b l o c k s ๐’ณ 1 ( โ‹… ) a n d ๐’ณ 2 ( โ‹… ) . ( 4 . 2 8 ) Suppose first that โ„ฐ ( ๐‘€ ) < 0 . Then the matrices ๐’ณ ๐‘— ( ๐‘ ) , ๐‘— โˆˆ { 1 , 2 } , are invertible. Indeed, if one of them is singular, then there exists a nonzero vector ๐‘ฃ โˆˆ โ„‚ ๐‘› such that ๐’ณ 1 ( ๐‘ ) ๐‘ฃ = 0 or ๐’ณ 2 ( ๐‘ ) ๐‘ฃ = 0 . Then ๐‘ฃ โˆ— โ„ฐ ( ๐‘€ ) ๐‘ฃ = ๐‘– ๐›ฟ ( ๐œ† ) ๐‘ฃ โˆ— ๐’ณ โˆ— ( ๐‘ ) ๐’ฅ ๐’ณ ( ๐‘ ) ๐‘ฃ = ๐‘– ๐›ฟ ( ๐œ† ) ๐‘ฃ โˆ— ๎€บ ๐’ณ โˆ— 1 ( ๐‘ ) ๐’ณ 2 ( ๐‘ ) โˆ’ ๐’ณ โˆ— 2 ( ๐‘ ) ๐’ณ 1 ๎€ป ( ๐‘ ) ๐‘ฃ = 0 , ( 4 . 2 9 ) which contradicts โ„ฐ ( ๐‘€ ) < 0 . Now we set ๐›ฝ 1 โˆถ = ๐ผ , ๐›ฝ 2 โˆถ = โˆ’ ๐’ณ 1 ( ๐‘ ) ๐’ณ 2 โˆ’ 1 ( ๐‘ ) , and ๐›ฝ โˆถ = ( ๐›ฝ 1 ๐›ฝ 2 ) . Then for this 2 ๐‘› ร— ๐‘› matrix ๐›ฝ we have ๐›ฝ ๐’ณ ( ๐‘ ) = 0 and, by a similar calculation as in (4.29), โ„ฐ ( ๐‘€ ) = ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— ( ๐‘ ) ๐’ฅ ๐’ณ ( ๐‘ ) = ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— 2 ๎€ท ๐›ฝ ( ๐‘ ) 2 ๐›ฝ โˆ— 1 โˆ’ ๐›ฝ 1 ๐›ฝ โˆ— 2 ๎€ธ ๐’ณ 2 ( ๐‘ ) = 2 ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— 2 ( ๎€ท ๐›ฝ ๐‘ ) I m 1 ๐›ฝ โˆ— 2 ๎€ธ ๐’ณ 2 ( ๐‘ ) = โˆ’ ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— 2 ( ๐‘ ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— ๐’ณ 2 ( ๐‘ ) , ( 4 . 3 0 ) where we used the equality ๐›ฝ ๐’ฅ ๐›ฝ โˆ— = 2 ๐‘– I m ( ๐›ฝ 1 ๐›ฝ โˆ— 2 ) . Since โ„ฐ ( ๐‘€ ) < 0 and ๐’ณ 2 ( ๐‘ ) is invertible, it follows that ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— > 0 . Conversely, assume that for a given matrix ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› there is ๐›ฝ = ( ๐›ฝ 1 ๐›ฝ 2 ) โˆˆ โ„‚ ๐‘› ร— 2 ๐‘› satisfying ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— > 0 and ๐›ฝ ๐’ณ ( ๐‘ ) = 0 . Condition ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— > 0 is equivalent to I m ( ๐›ฝ 1 ๐›ฝ โˆ— 2 ) < 0 when I m ( ๐œ† ) > 0 and to I m ( ๐›ฝ 1 ๐›ฝ โˆ— 2 ) > 0 when I m ( ๐œ† ) < 0 . The positive or negative definiteness of I m ( ๐›ฝ 1 ๐›ฝ โˆ— 2 ) implies the invertibility of ๐›ฝ 1 and ๐›ฝ 2 ; see Remark 2.2. Therefore, from the equality ๐›ฝ 1 ๐’ณ 1 ( ๐‘ ) + ๐›ฝ 2 ๐’ณ 2 ( ๐‘ ) = ๐›ฝ ๐’ณ ( ๐‘ ) = 0 , we obtain ๐’ณ 1 ( ๐‘ ) = โˆ’ ๐›ฝ 1 โˆ’ 1 ๐›ฝ 2 ๐’ณ 2 ( ๐‘ ) , and so โ„ฐ ๎€บ ๐’ณ ( ๐‘€ ) = ๐‘– ๐›ฟ ( ๐œ† ) โˆ— 1 ( ๐‘ ) ๐’ณ 2 ( ๐‘ ) โˆ’ ๐’ณ โˆ— 2 ( ๐‘ ) ๐’ณ 1 ๎€ป ( ๐‘ ) = ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— 2 ( ๐‘ ) ๐›ฝ 1 โˆ’ 1 ๎€ท ๐›ฝ 2 ๐›ฝ โˆ— 1 โˆ’ ๐›ฝ 1 ๐›ฝ โˆ— 2 ๎€ธ ๐›ฝ 1 โˆ— โˆ’ 1 ๐’ณ 2 ( ๐‘ ) = โˆ’ ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— 2 ( ๐‘ ) ๐›ฝ 1 โˆ’ 1 ๐›ฝ ๐’ฅ ๐›ฝ โˆ— ๐›ฝ 1 โˆ— โˆ’ 1 ๐’ณ 2 ( ๐‘ ) . ( 4 . 3 1 ) The matrix ๐’ณ 2 ( ๐‘ ) is invertible, because if ๐’ณ 2 ( ๐‘ ) ๐‘‘ = 0 for some nonzero vector ๐‘‘ โˆˆ โ„‚ ๐‘› , then ๐’ณ 1 ( ๐‘ ) ๐‘‘ = โˆ’ ๐›ฝ 1 โˆ’ 1 ๐›ฝ 2 ๐’ณ 2 ( ๐‘ ) ๐‘‘ = 0 , showing that r a n k ๐’ณ ( ๐‘ ) < ๐‘› . This however contradicts r a n k ๐’ณ ( ๐‘ ) = ๐‘› which we have from the definition of the Weyl solution ๐’ณ ( โ‹… ) in (4.16). Consequently, (4.31) yields through ๐‘– ๐›ฟ ( ๐œ† ) ๐›ฝ ๐’ฅ ๐›ฝ โˆ— > 0 that โ„ฐ ( ๐‘€ ) < 0 .
If the matrix ๐›ฝ does not satisfy ๐›ฝ ๐›ฝ โˆ— = ๐ผ , then we modify it according to the procedure described in the proof of Theorem 4.10. Finally, since ๐œ† โˆˆ โ„‚ โงต โ„ , we get from Proposition 4.3(iii) and Lemma 4.4 that the matrix ๐›ฝ ๎‚ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible which in turn implies through the calculation in (4.27) that ๎‚ ๐‘€ = โˆ’ [ ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) ] โˆ’ 1 ๐›ฝ ๐‘ ( ๐‘ , ๐œ† , ๐›ผ ) = ๐‘€ ( ๐œ† ) .

In the following lemma we derive some additional properties of the Weyl disk and the ๐‘€ ( ๐œ† ) -function. Special cases of this statement can be found in [8, Lemma  2.9], [33, Theorem  3.1], [9, Lemma  2.14], [14, Lemma  3.2(ii)], [1, Theorem  3.7], [2, Lemma  3.7], and [3, Theorem  4.13].

Theorem 4.13. Let ๐›ผ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . For any matrix ๐‘€ โˆˆ ๐ท ( ๐œ† ) we have | | | | ๎€œ ๐›ฟ ( ๐œ† ) I m ( ๐‘€ ) โ‰ฅ I m ( ๐œ† ) ๐‘ ๐‘Ž ๐’ณ ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฒ ( ๐‘ก ) ๐’ณ ๐œŽ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ฮ” ๐‘ก โ‰ฅ 0 . ( 4 . 3 2 ) In addition, under Hypothesis 4.2, we have ๐›ฟ ( ๐œ† ) I m ( ๐‘€ ) > 0 .

Proof. By identity (4.18), for any matrix ๐‘€ โˆˆ ๐ท ( ๐œ† ) , we have | | | | ๎€œ 2 ๐›ฟ ( ๐œ† ) I m ( ๐‘€ ) = โˆ’ โ„ฐ ( ๐‘€ ) + 2 I m ( ๐œ† ) ๐‘ ๐‘Ž ๐’ณ ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฒ ( ๐‘ก ) ๐’ณ ๐œŽ | | | | ๎€œ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ฮ” ๐‘ก โ‰ฅ 2 I m ( ๐œ† ) ๐‘ ๐‘Ž ๐’ณ ๐œŽ โˆ— ( ๎‚‹ ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฒ ( ๐‘ก ) ๐’ณ ๐œŽ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ฮ” ๐‘ก , ( 4 . 3 3 ) which yields together with ๎‚‹ ๐’ฒ ( ๐‘ก ) โ‰ฅ 0 on [ ๐‘Ž , ๐œŒ ( ๐‘ ) ] ๐•‹ the inequalities in (4.32). The last assertion in Theorem 4.13 is a simple consequence of Hypothesis 4.2.

In the last part of this section we wish to study the effect of changing ๐›ผ , which is one of the parameters of the ๐‘€ ( ๐œ† ) -function and the Weyl solution ๐’ณ ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ) , when ๐›ผ varies within the set ฮ“ . For this purpose we will use the ๐‘€ ( ๐œ† ) -function with all its arguments in the following two statements.

Lemma 4.14. Let ๐›ผ , ๐›ฝ , ๐›พ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . Then ๎€บ ๐‘€ ( ๐œ† , ๐‘ , ๐›ผ , ๐›ฝ ) = ๐›ผ ๐’ฅ ๐›พ โˆ— + ๐›ผ ๐›พ โˆ— ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ๎€ป ๎€บ ๐›ผ ๐›พ โˆ— โˆ’ ๐›ผ ๐’ฅ ๐›พ โˆ— ๎€ป ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) โˆ’ 1 . ( 4 . 3 4 )

Proof. Let ๐‘€ ( ๐‘ , ๐œ† , ๐›ผ , ๐›ฝ ) and ๐‘€ ( ๐‘ , ๐œ† , ๐›พ , ๐›ฝ ) be given via (4.13), and consider the Weyl solutions ๐’ณ ๐›ผ ( โ‹… ) โˆถ = ๐’ณ ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ( ๐‘ , ๐œ† , ๐›ผ , ๐›ฝ ) ) and ๐’ณ ๐›พ ( โ‹… ) โˆถ = ๐’ณ ( โ‹… , ๐œ† , ๐›พ , ๐‘€ ( ๐‘ , ๐œ† , ๐›พ , ๐›ฝ ) ) defined by (4.16) with ๐‘€ = ๐‘€ ( ๐‘ , ๐œ† , ๐›ผ , ๐›ฝ ) and ๐‘€ = ๐‘€ ( ๐‘ , ๐œ† , ๐›พ , ๐›ฝ ) , respectively. First we prove that the two Weyl solutions ๐’ณ ๐›ผ ( โ‹… ) and ๐’ณ ๐›พ ( โ‹… ) differ by a constant nonsingular multiple. By definition, ๐›ฝ ๐’ณ ๐›ผ ( ๐‘ ) = 0 and ๐›ฝ ๐’ณ ๐›พ ( ๐‘ ) = 0 , which implies through (4.3) that ๐’ณ ๐›ผ ( ๐‘ ) = ๐’ฅ ๐›ฝ โˆ— ๐‘ƒ ๐›ผ and ๐’ณ ๐›พ ( ๐‘ ) = ๐’ฅ ๐›ฝ โˆ— ๐‘ƒ ๐›พ for some matrices ๐‘ƒ ๐›ผ , ๐‘ƒ ๐›พ โˆˆ โ„‚ ๐‘› ร— ๐‘› , which are invertible by Remark 4.11. This implies that ๐’ณ ๐›ผ ( ๐‘ ) ๐‘ƒ ๐›ผ โˆ’ 1 = ๐’ฅ ๐›ฝ โˆ— = ๐’ณ ๐›พ ( ๐‘ ) ๐‘ƒ ๐›พ โˆ’ 1 . Consequently, ๐’ณ ๐›ผ ( ๐‘ ) = ๐’ณ ๐›พ ( ๐‘ ) ๐‘ƒ , where ๐‘ƒ โˆถ = ๐‘ƒ ๐›พ โˆ’ 1 ๐‘ƒ ๐›ผ . By the uniqueness of solutions of system ( ๐’ฎ ๐œ† ), see Theorem 3.4, we obtain that ๐’ณ ๐›ผ ( โ‹… ) = ๐’ณ ๐›พ ( โ‹… ) ๐‘ƒ on [ ๐‘Ž , ๐‘ ] ๐•‹ . Upon the evaluation at ๐‘ก = ๐‘Ž we get ๎‚ต ๐ผ ๎‚ถ ๎‚ต ๐ผ ๎‚ถ ฮจ ( ๐‘Ž , ๐œ† , ๐›ผ ) ๐‘€ ( ๐œ† , ๐‘ , ๐›ผ , ๐›ฝ ) = ฮจ ( ๐‘Ž , ๐œ† , ๐›พ ) ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ๐‘ƒ . ( 4 . 3 5 ) Since the matrices ฮจ ( ๐‘Ž , ๐œ† , ๐›ผ ) = ( ๐›ผ โˆ— โˆ’ ๐’ฅ ๐›ผ โˆ— ) and ฮจ ( ๐‘Ž , ๐œ† , ๐›พ ) = ( ๐›พ โˆ— โˆ’ ๐’ฅ ๐›พ โˆ— ) are unitary, it follows from (4.35) that ๎‚ต ๐ผ ๎‚ถ = ๎‚ต ๐›ผ ๎‚ถ ๎€ท ๐›พ ๐‘€ ( ๐œ† , ๐‘ , ๐›ผ , ๐›ฝ ) ๐›ผ ๐’ฅ โˆ— โˆ’ ๐’ฅ ๐›พ โˆ— ๎€ธ ๎‚ต ๐ผ ๎‚ถ ๐‘ƒ = ๎‚ต ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ๐›ผ ๐›พ โˆ— โˆ’ ๐›ผ ๐’ฅ ๐›พ โˆ— ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ๐›ผ ๐’ฅ ๐›พ โˆ— + ๐›ผ ๐›พ โˆ— ๎‚ถ ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ๐‘ƒ . ( 4 . 3 6 ) The first row above yields that ๐‘ƒ = [ ๐›ผ ๐›พ โˆ— โˆ’ ๐›ผ ๐’ฅ ๐›พ โˆ— ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ] โˆ’ 1 , while the second row is then written as identity (4.34).

Corollary 4.15. Let ๐›ผ , ๐›ฝ , ๐›พ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . With notation (4.16) and (4.13) we have ๎€บ ๐’ณ ( โ‹… , ๐œ† , ๐›ผ , ๐‘€ ( ๐œ† , ๐‘ , ๐›ผ , ๐›ฝ ) ) = ๐’ณ ( โ‹… , ๐œ† , ๐›พ , ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) ) ๐›ผ ๐›พ โˆ— โˆ’ ๐›ผ ๐’ฅ ๐›พ โˆ— ๎€ป ๐‘€ ( ๐œ† , ๐‘ , ๐›พ , ๐›ฝ ) โˆ’ 1 . ( 4 . 3 7 )

Proof. The above identity follows from (4.35) and the formula for the matrix ๐‘ƒ from the end of the proof of Lemma 4.14.

5. Geometric Properties of Weyl Disks

In this section we study the geometric properties of the Weyl disks as the point ๐‘ moves through the interval [ ๐‘Ž , โˆž ) ๐•‹ . Our first result shows that the Weyl disks ๐ท ( ๐œ† , ๐‘ ) are nested. This statement generalizes the results in [11, Theorem  4.5], [66, Section  3.2.1], [9, equation  (2.70)], [14, Theorem  3.1], [3, Theorem  4.4], and [5, Theorem  3.3(i)].

Theorem 5.1 (nesting property of Weyl disks). Let ๐›ผ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . Then ๐ท ๎€ท ๐œ† , ๐‘ 2 ๎€ธ ๎€ท โŠ† ๐ท ๐œ† , ๐‘ 1 ๎€ธ , f o r e v e r y ๐‘ 1 , ๐‘ 2 โˆˆ [ ) ๐‘Ž , โˆž ๐•‹ , ๐‘ 1 < ๐‘ 2 . ( 5 . 1 )

Proof. Let ๐‘ 1 , ๐‘ 2 โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ with ๐‘ 1 < ๐‘ 2 , and take ๐‘€ โˆˆ ๐ท ( ๐œ† , ๐‘ 2 ) , that is, โ„ฐ ( ๐‘€ , ๐‘ 2 ) โ‰ค 0 . From identity (4.18) with ๐‘ = ๐‘ 1 and later with ๐‘ = ๐‘ 2 and by using ๎‚‹ ๐’ฒ ( โ‹… ) โ‰ฅ 0 , we have โ„ฐ ๎€ท ๐‘€ , ๐‘ 1 ๎€ธ ( 4 . 1 8 ) | | | | ๎€œ = โˆ’ 2 ๐›ฟ ( ๐œ† ) I m ( ๐‘€ ) + 2 I m ( ๐œ† ) ๐‘ 1 ๐‘Ž ๐’ณ ๐œŽ โˆ— ๎‚‹ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฒ ( ๐‘ก ) ๐’ณ ๐œŽ | | | | ๎€œ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ฮ” ๐‘ก โ‰ค โˆ’ 2 ๐›ฟ ( ๐œ† ) I m ( ๐‘€ ) + 2 I m ( ๐œ† ) ๐‘ 2 ๐‘Ž ๐’ณ ๐œŽ โˆ— ( ๎‚‹ ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฒ ( ๐‘ก ) ๐’ณ ๐œŽ ( ๐‘ก , ๐œ† , ๐›ผ , ๐‘€ ) ฮ” ๐‘ก ( 4 . 1 8 ) ๎€ท = โ„ฐ ๐‘€ , ๐‘ 2 ๎€ธ โ‰ค 0 . ( 5 . 2 ) Therefore, by Definition 4.9, the matrix ๐‘€ belongs to ๐ท ( ๐œ† , ๐‘ 1 ) , which shows the result.

Similarly for the regular case (Hypothesis 4.2) we now introduce the following assumption.

Hypothesis 5.2. There exists ๐‘ 0 โˆˆ ( ๐‘Ž , โˆž ) ๐•‹ such that Hypothesis 4.2 is satisfied with ๐‘ = ๐‘ 0 ; that is, inequality (4.8) holds with ๐‘ = ๐‘ 0 for every ๐œ† โˆˆ โ„‚ .

From Hypothesis 5.2 it follows by ๎‚‹ ๐’ฒ ( โ‹… ) โ‰ฅ 0 that inequality (4.8) holds for every ๐‘ โˆˆ [ ๐‘ 0 , โˆž ) ๐•‹ .

For the study of the geometric properties of Weyl disks we will use the following representation: โ„ฐ ( ๐‘€ , ๐‘ ) = ๐‘– ๐›ฟ ( ๐œ† ) ๐’ณ โˆ— ๎€ท ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) ๐’ฅ ๐’ณ ( ๐‘ , ๐œ† , ๐›ผ , ๐‘€ ) = ๐ผ ๐‘€ โˆ— ๎€ธ ๎‚ต โ„ฑ ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ข โˆ— ๐ผ ๐‘€ ๎‚ถ ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ข ( ๐‘ , ๐œ† , ๐›ผ ) โ„‹ ( ๐‘ , ๐œ† , ๐›ผ ) ๎‚ถ ๎‚ต , ( 5 . 3 ) of the matrix โ„ฐ ( ๐‘€ , ๐‘ ) , where we define on [ ๐‘Ž , โˆž ) ๐•‹ the ๐‘› ร— ๐‘› matrices โ„ฑ ( โ‹… , ๐œ† , ๐›ผ ) โˆถ = ๐‘– ๐›ฟ ( ๐œ† ) ๐‘ โˆ— ๎‚ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) ๐’ฅ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) , ๐’ข ( โ‹… , ๐œ† , ๐›ผ ) โˆถ = ๐‘– ๐›ฟ ( ๐œ† ) โˆ— ๎‚ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) ๐’ฅ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) , โ„‹ ( โ‹… , ๐œ† , ๐›ผ ) โˆถ = ๐‘– ๐›ฟ ( ๐œ† ) โˆ— ๎‚ ( โ‹… , ๐œ† , ๐›ผ ) ๐’ฅ ๐‘ ( โ‹… , ๐œ† , ๐›ผ ) . ( 5 . 4 ) Since โ„ฐ ( ๐‘€ , ๐‘ ) is Hermitian, it follows that โ„ฑ ( โ‹… , ๐œ† , ๐›ผ ) and โ„‹ ( โ‹… , ๐œ† , ๐›ผ ) are also Hermitian. Moreover, by (4.7), we have โ„‹ ( ๐‘Ž , ๐œ† , ๐›ผ ) = 0 . In addition, if ๐‘ โˆˆ [ ๐‘ 0 , โˆž ) ๐•‹ , then Corollary 3.7 and Hypothesis 5.2 yield for any ๐œ† โˆˆ โ„‚ โงต โ„ | | | | ๎€œ โ„‹ ( ๐‘ , ๐œ† , ๐›ผ ) = 2 I m ( ๐œ† ) ๐‘ ๐‘Ž ๎‚ ๐‘ ๐œŽ โˆ— ๎‚‹ ๎‚ ๐‘ ( ๐‘ก , ๐œ† , ๐›ผ ) ๐’ฒ ( ๐‘ก ) ๐œŽ ( ๐‘ก , ๐œ† , ๐›ผ ) ฮ” ๐‘ก > 0 . ( 5 . 5 ) Therefore, โ„‹ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible (positive definite) for all ๐‘ โˆˆ [ ๐‘ 0 , โˆž ) ๐•‹ and monotone nondecreasing as ๐‘ โ†’ โˆž , with a consequence that โ„‹ โˆ’ 1 ( ๐‘ , ๐œ† , ๐›ผ ) is monotone nonincreasing as ๐‘ โ†’ โˆž . The following factorization of โ„ฐ ( ๐‘€ , ๐‘ ) holds true; see also [2, equation  (4.11)].

Lemma 5.3. Let ๐›ผ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . With the notation (5.4), for any ๐‘€ โˆˆ โ„‚ ๐‘› ร— ๐‘› and ๐‘ โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ we have โ„ฐ ( ๐‘€ , ๐‘ ) = โ„ฑ ( ๐‘ , ๐œ† , ๐›ผ ) โˆ’ ๐’ข โˆ— ( ๐‘ , ๐œ† , ๐›ผ ) โ„‹ โˆ’ 1 + ๎€บ ๐’ข ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ข ( ๐‘ , ๐œ† , ๐›ผ ) โˆ— ( ๐‘ , ๐œ† , ๐›ผ ) โ„‹ โˆ’ 1 ( ๐‘ , ๐œ† , ๐›ผ ) + ๐‘€ โˆ— ๎€ป โ„‹ ๎€บ โ„‹ ( ๐‘ , ๐œ† , ๐›ผ ) โˆ’ 1 ๎€ป , ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ข ( ๐‘ , ๐œ† , ๐›ผ ) + ๐‘€ ( 5 . 6 ) whenever the matrix โ„‹ ( ๐‘ , ๐œ† , ๐›ผ ) is invertible.

Proof. The result is shown by a direct calculation.

The following identity is a generalization of its corresponding versions in [11, Lemma  4.3], [1, Lemma  3.3], [14, Proposition  3.2], [2, Lemma  4.2], [3, Lemma  4.6], and [5, Theorem  5.6].

Lemma 5.4. Let ๐›ผ โˆˆ ฮ“ and ๐œ† โˆˆ โ„‚ โงต โ„ . With the notation (5.4), for any ๐‘ โˆˆ [ ๐‘Ž , โˆž ) ๐•‹ , we have ๐’ข โˆ— ( ๐‘ , ๐œ† , ๐›ผ ) โ„‹ โˆ’ 1 ( ๐‘ , ๐œ† , ๐›ผ ) ๐’ข ( ๐‘ , ๐œ† , ๐›ผ ) โˆ’ โ„ฑ ( ๐‘ , ๐œ† , ๐›ผ ) = โ„‹ โˆ’ 1 ๎‚€ ๐‘ , ๎‚ ๐œ† , ๐›ผ , ( 5 . 7 ) whenever the matrices โ„‹