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International Journal of Mathematics and Mathematical Sciences
Volumeย 2011ย (2011), Article IDย 161209, 10 pages
Research Article

Fine Spectra of Tridiagonal Symmetric Matrices

Faculty of Arts and Sciences, Adฤฑyaman University, 02040 Adฤฑyaman, Turkey

Received 14 December 2010; Accepted 10 March 2011

Academic Editor: Martinย Bohner

Copyright ยฉ 2011 Muhammed Altun. 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.


The fine spectra of upper and lower triangular banded matrices were examined by several authors. Here we determine the fine spectra of tridiagonal symmetric infinite matrices and also give the explicit form of the resolvent operator for the sequence spaces ๐‘0, ๐‘, โ„“1, and โ„“โˆž.

1. Introduction

The spectrum of an operator is a generalization of the notion of eigenvalues for matrices. The spectrum over a Banach space is partitioned into three parts, which are the point spectrum, the continuous spectrum, and the residual spectrum. The calculation of these three parts of the spectrum of an operator is called the fine spectrum of the operator.

The spectrum and fine spectrum of linear operators defined by some particular limitation matrices over some sequence spaces was studied by several authors. We introduce the knowledge in the existing literature concerning the spectrum and the fine spectrum. Wenger [1] examined the fine spectrum of the integer power of the Cesร ro operator over ๐‘ and, Rhoades [2] generalized this result to the weighted mean methods. Reade [3] worked on the spectrum of the Cesร ro operator over the sequence space ๐‘0. Gonzรกles [4] studied the fine spectrum of the Cesร ro operator over the sequence space โ„“๐‘. Okutoyi [5] computed the spectrum of the Cesร ro operator over the sequence space ๐‘๐‘ฃ. Recently, Rhoades and Yildirim [6] examined the fine spectrum of factorable matrices over ๐‘0 and ๐‘. CoลŸkun [7] studied the spectrum and fine spectrum for the p-Cesร ro operator acting over the space ๐‘0. Akhmedov and BaลŸar [8, 9] have determined the fine spectrum of the Cesร ro operator over the sequence spaces ๐‘0, โ„“โˆž, and โ„“๐‘. In a recent paper, Furkan, et al. [10] determined the fine spectrum of ๐ต(๐‘Ÿ,๐‘ ,๐‘ก) over the sequence spaces ๐‘0 and ๐‘, where ๐ต(๐‘Ÿ,๐‘ ,๐‘ก) is a lower triangular triple-band matrix. Later, Altun and Karakaya [11] computed the fine spectra for Lacunary matrices over ๐‘0 and ๐‘.

In this work, our purpose is to determine the fine spectra of the operator, for which the corresponding matrix is a tridiagonal symmetric matrix, over the sequence spaces ๐‘0, ๐‘, โ„“1, and โ„“โˆž. Also we will give the explicit form of the resolvent for this operator and compute the norm of the resolvent operator when it exists and is continuous.

Let ๐‘‹ and ๐‘Œ be Banach spaces and ๐‘‡โˆถ๐‘‹โ†’๐‘Œ be a bounded linear operator. By โ„›(๐‘‡), we denote the range of ๐‘‡, that is, โ„›(๐‘‡)={๐‘ฆโˆˆ๐‘Œโˆถ๐‘ฆ=๐‘‡๐‘ฅ;๐‘ฅโˆˆ๐‘‹}.(1.1) By ๐ต(๐‘‹), we denote the set of all bounded linear operators on ๐‘‹ into itself. If ๐‘‹ is any Banach space, and let ๐‘‡โˆˆ๐ต(๐‘‹) then the adjoint ๐‘‡โˆ— of ๐‘‡ is a bounded linear operator on the dual ๐‘‹โˆ— of ๐‘‹ defined by (๐‘‡โˆ—๐œ™)(๐‘ฅ)=๐œ™(๐‘‡๐‘ฅ) for all ๐œ™โˆˆ๐‘‹โˆ— and ๐‘ฅโˆˆ๐‘‹. Let ๐‘‹โ‰ {๐œƒ} be a complex normed space and ๐‘‡โˆถ๐’Ÿ(๐‘‡)โ†’๐‘‹ be a linear operator with domain ๐’Ÿ(๐‘‡)โŠ‚๐‘‹. With ๐‘‡, we associate the operator ๐‘‡๐œ†=๐‘‡โˆ’๐œ†๐ผ,(1.2) where ๐œ† is a complex number and ๐ผ is the identity operator on ๐’Ÿ(๐‘‡). If ๐‘‡๐œ† has an inverse, which is linear, we denote it by ๐‘‡๐œ†โˆ’1, that is ๐‘‡๐œ†โˆ’1=(๐‘‡โˆ’๐œ†๐ผ)โˆ’1(1.3) and call it the resolvent operator of ๐‘‡๐œ†. If ๐œ†=0, we will simply write ๐‘‡โˆ’1. Many properties of ๐‘‡๐œ† and ๐‘‡๐œ†โˆ’1 depend on ๐œ†, and spectral theory is concerned with those properties. For instance, we will be interested in the set of all ๐œ† in the complex plane such that ๐‘‡๐œ†โˆ’1 exists. Boundedness of ๐‘‡๐œ†โˆ’1 is another property that will be essential. We shall also ask for what ๐œ†๎…žs the domain of ๐‘‡๐œ†โˆ’1 is dense in ๐‘‹. For our investigation of ๐‘‡, ๐‘‡๐œ†, and ๐‘‡๐œ†โˆ’1, we need some basic concepts in spectral theory which are given as follows (see [12, pages 370-371]).

Let ๐‘‹โ‰ {๐œƒ} be a complex normed space, and let ๐‘‡โˆถ๐’Ÿ(๐‘‡)โ†’๐‘‹ be a linear operator with domain ๐’Ÿ(๐‘‡)โŠ‚๐‘‹. A regular value ๐œ† of ๐‘‡ is a complex number such that (R1)๐‘‡๐œ†โˆ’1 exists,(R2)๐‘‡๐œ†โˆ’1 is bounded, and(R3)๐‘‡๐œ†โˆ’1 is defined on a set which is dense in ๐‘‹.

The resolvent set ๐œŒ(๐‘‡) of ๐‘‡ is the set of all regular values ๐œ† of ๐‘‡. Its complement ๐œŽ(๐‘‡)=โ„‚โงต๐œŒ(๐‘‡) in the complex plane โ„‚ is called the spectrum of ๐‘‡. Furthermore, the spectrum ๐œŽ(๐‘‡) is partitioned into three disjoint sets as follows: the point spectrum ๐œŽ๐‘(๐‘‡) is the set such that ๐‘‡๐œ†โˆ’1 does not exist. A ๐œ†โˆˆ๐œŽ๐‘(๐‘‡) is called an eigenvalue of ๐‘‡. The continuous spectrum ๐œŽ๐‘(๐‘‡) is the set such that ๐‘‡๐œ†โˆ’1 exists and satisfies (R3) but not (R2). The residual spectrum ๐œŽ๐‘Ÿ(๐‘‡) is the set such that ๐‘‡๐œ†โˆ’1 exists but does not satisfy (R3).

A triangle is a lower triangular matrix with all of the principal diagonal elements nonzero. We shall write โ„“โˆž, ๐‘, and ๐‘0 for the spaces of all bounded, convergent, and null sequences, respectively. And by โ„“๐‘, we denote the space of all ๐‘-absolutely summable sequences, where 1โ‰ค๐‘<โˆž. Let ๐œ‡ and ๐›พ be two sequence spaces and ๐ด=(๐‘Ž๐‘›๐‘˜) be an infinite matrix of real or complex numbers ๐‘Ž๐‘›๐‘˜, where ๐‘›,๐‘˜โˆˆโ„•. Then, we say that ๐ด defines a matrix mapping from ๐œ‡ into ๐›พ, and we denote it by writing ๐ดโˆถ๐œ‡โ†’๐›พ, if for every sequence ๐‘ฅ=(๐‘ฅ๐‘˜)โˆˆ๐œ‡ the sequence ๐ด๐‘ฅ={(๐ด๐‘ฅ)๐‘›}, the ๐ด-transform of ๐‘ฅ, is in ๐›พ, where(๐ด๐‘ฅ)๐‘›=๎“๐‘˜๐‘Ž๐‘›๐‘˜๐‘ฅ๐‘˜(๐‘›โˆˆโ„•).(1.4) By (๐œ‡โˆถ๐›พ), we denote the class of all matrices ๐ด such that ๐ดโˆถ๐œ‡โ†’๐›พ. Thus, ๐ดโˆˆ(๐œ‡โˆถ๐›พ) if and only if the series on the right side of (1.4) converges for each ๐‘›โˆˆโ„• and every ๐‘ฅโˆˆ๐œ‡, and we have ๐ด๐‘ฅ={(๐ด๐‘ฅ)๐‘›}๐‘›โˆˆโ„•โˆˆ๐›พ for all ๐‘ฅโˆˆ๐œ‡.

A tridiagonal symmetric infinite matrix is of the form โŽกโŽขโŽขโŽขโŽขโŽขโŽฃโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ๐‘†=๐‘†(๐‘ž,๐‘Ÿ)=๐‘ž๐‘Ÿ0000โ‹ฏ๐‘Ÿ๐‘ž๐‘Ÿ000โ‹ฏ0๐‘Ÿ๐‘ž๐‘Ÿ00โ‹ฏ00๐‘Ÿ๐‘ž๐‘Ÿ0โ‹ฏโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฎโ‹ฑ,(1.5) where ๐‘ž,๐‘Ÿโˆˆโ„‚. The spectral results are clear when ๐‘Ÿ=0, so for the sequel we will have ๐‘Ÿโ‰ 0.

Theorem 1.1 (cf. [13]). Let ๐‘‡ be an operator with the associated matrix ๐ด=(๐‘Ž๐‘›๐‘˜). (i)๐‘‡โˆˆ๐ต(๐‘) if and only if โ€–๐ดโ€–โˆถ=sup๐‘›โˆž๎“๐‘˜=1||๐‘Ž๐‘›๐‘˜||๐‘Ž<โˆž,(1.6)๐‘˜โˆถ=lim๐‘›โ†’โˆž๐‘Ž๐‘›๐‘˜existsforeach๐‘˜,(1.7)๐‘Žโˆถ=limโˆž๐‘›โ†’โˆž๎“๐‘˜=1๐‘Ž๐‘›๐‘˜exists.(1.8)(ii)๐‘‡โˆˆ๐ต(๐‘0) if and only if (1.6) and (1.7) with ๐‘Ž๐‘˜=0 for each ๐‘˜.(iii)๐‘‡โˆˆ๐ต(โ„“โˆž) if and only if (1.6). In these cases, the operator norm of ๐‘‡ is โ€–๐‘‡โ€–(โ„“โˆžโˆถโ„“โˆž)=โ€–๐‘‡โ€–(๐‘โˆถ๐‘)=โ€–๐‘‡โ€–(๐‘0โˆถ๐‘0)=โ€–๐ดโ€–.(1.9)(iv)๐‘‡โˆˆ๐ต(โ„“1) if and only if โ€–โ€–๐ด๐‘กโ€–โ€–=sup๐‘˜โˆž๎“๐‘›=1||๐‘Ž๐‘›๐‘˜||<โˆž.(1.10) In this case, the operator norm of ๐‘‡ is โ€–๐‘‡โ€–(โ„“1โˆถโ„“1)=โ€–๐ด๐‘กโ€–.

Corollary 1.2. Let ๐œ‡โˆˆ{๐‘0,๐‘,โ„“1,โ„“โˆž}. ๐‘†(๐‘ž,๐‘Ÿ)โˆถ๐œ‡โ†’๐œ‡ is a bounded linear operator and โ€–๐‘†(๐‘ž,๐‘Ÿ)โ€–(๐œ‡โˆถ๐œ‡)=|๐‘ž|+2|๐‘Ÿ|.

2. The Spectra and Point Spectra

Theorem 2.1. ๐œŽ๐‘(๐‘†,๐œ‡)=โˆ… for ๐œ‡โˆˆ{โ„“1,๐‘0,๐‘}.

Proof. Since โ„“1โŠ‚๐‘0โŠ‚๐‘, it is enough to show that ๐œŽ๐‘(๐‘†,๐‘)=โˆ…. Let ๐œ† be an eigenvalue of the operator ๐‘†. An eigenvector ๐‘ฅ=(๐‘ฅ0,๐‘ฅ1,โ€ฆ)โˆˆ๐‘ corresponding to this eigenvalue satisfies the linear system of equations: ๐‘ž๐‘ฅ0+๐‘Ÿ๐‘ฅ1=๐œ†๐‘ฅ0๐‘Ÿ๐‘ฅ0+๐‘ž๐‘ฅ1+๐‘Ÿ๐‘ฅ2=๐œ†๐‘ฅ1๐‘Ÿ๐‘ฅ1+๐‘ž๐‘ฅ2+๐‘Ÿ๐‘ฅ3=๐œ†๐‘ฅ2โ‹ฎ.(2.1) If ๐‘ฅ0=0, then ๐‘ฅ๐‘˜=0 for all ๐‘˜โˆˆโ„•. Hence ๐‘ฅ0โ‰ 0. Without loss of generality we can suppose ๐‘ฅ0=1. Then ๐‘ฅ1=(๐œ†โˆ’๐‘ž)/๐‘Ÿ and the system of equations turn into the linear homogeneous recurrence relation ๐‘ฅ๐‘›+๐‘๐‘ฅ๐‘›โˆ’1+๐‘ฅ๐‘›โˆ’2=0for๐‘›โ‰ฅ2,(2.2) where ๐‘=(๐‘žโˆ’๐œ†)/๐‘Ÿ. The characteristic polynomial of the recurrence relation is ๐‘ฅ2+๐‘๐‘ฅ+1=0.(2.3) There are three cases here.Case 1 (๐‘=โˆ’2). Then characteristic polynomial has only one root: ๐›ผ=1. Hence, the solution of the recurrence relation is of the form ๐‘ฅ๐‘›=(๐ด+๐ต๐‘›)(๐›ผ)๐‘›=๐ด+๐ต๐‘›,(2.4) where ๐ด and ๐ต are constants which can be determined by the first two terms ๐‘ฅ0 and ๐‘ฅ1. 1=๐‘ฅ0=๐ด+๐ต0, so we have ๐ด=1. And โˆ’๐‘=๐‘ฅ1=๐ด+๐ต1, so we have ๐ต=1. Then ๐‘ฅ๐‘›=๐‘›+1. This means (๐‘ฅ๐‘›)โˆ‰๐‘. So, we conclude that there is no eigenvalue in this case.Case 2 (๐‘=2). Then characteristic polynomial has only one root: ๐›ผ=โˆ’1. The solution of the recurrence relation, found as in Case 1, is ๐‘ฅ๐‘›=(๐‘›+1)(โˆ’1)๐‘›. So, there is no eigenvalue in this case.Case 3 (๐‘โ‰ ยฑ2). Then the characteristic polynomial has two distinct roots ๐›ผ1โ‰ ยฑ1 and ๐›ผ2โ‰ ยฑ1 with ๐›ผ1๐›ผ2=1. Let |๐›ผ1|โ‰ฅ1โ‰ฅ|๐›ผ2|. The solution of the recurrence relation is of the form ๐‘ฅ๐‘›๎€ท๐›ผ=๐ด1๎€ธ๐‘›๎€ท๐›ผ+๐ต2๎€ธ๐‘›.(2.5) Using the first two terms and the fact that ๐‘=โˆ’(๐›ผ1+๐›ผ2), we get ๐ด=๐›ผ1/(๐›ผ1โˆ’๐›ผ2) and ๐ต=๐›ผ2/(๐›ผ2โˆ’๐›ผ1). So we have ๐‘ฅ๐‘›=๐›ผ1๐‘›+1โˆ’๐›ผ2๐‘›+1๐›ผ1โˆ’๐›ผ2.(2.6) If |๐›ผ1|>1>|๐›ผ2|, then ||๐‘ฅ๐‘›||โ‰ฅ1||๐›ผ1โˆ’๐›ผ2||๎‚€||๐›ผ1||๐‘›+1โˆ’||๐›ผ2||๐‘›+1๎‚.(2.7)So lim๐‘›|๐‘ฅ๐‘›|=โˆž, which means (๐‘ฅ๐‘›)โˆ‰๐‘. Now, if |๐›ผ1|=|๐›ผ2|=1, then there exists ๐œƒโˆˆ(0,๐œ‹) such that ๐›ผ1=๐‘’๐‘–๐œƒ and ๐›ผ2=๐‘’โˆ’๐‘–๐œƒ. So, ๐‘ฅ๐‘›=[sin(๐‘›+1)๐œƒ]/sin๐œƒ. Again we have (๐‘ฅ๐‘›)โˆ‰๐‘. Hence there is no eigenvalue also in this case.

Repeating all the steps in the proof of this theorem for โ„“โˆž, we get to the following.

Theorem 2.2. ๐œŽ๐‘(๐‘†,โ„“โˆž)=(๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ).

Theorem 2.3. Let ๐‘=(๐‘žโˆ’๐œ†)/๐‘Ÿ. Let ๐›ผ1 and ๐›ผ2 be the roots of the polynomial ๐‘ƒ(๐‘ฅ)=๐‘ฅ2+๐‘๐‘ฅ+1, with |๐›ผ2|>1>|๐›ผ1|. Then the resolvent operator over ๐‘0 is ๐‘†๐œ†โˆ’1=(๐‘ ๐‘›๐‘˜), where ๐‘ ๐‘›๐‘˜=1๐‘Ÿ๎€ท๐›ผ21๎€ธโ‹…๎ƒฏ๐›ผโˆ’11๐‘›โˆ’๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3๐›ผif๐‘›โ‰ฅ๐‘˜1โˆ’๐‘›+๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3if๐‘›<๐‘˜.(2.8) Moreover, this operator is continuous and the domain of the operator is the whole space ๐‘0.

Proof. Let ๐›ผ1 and ๐›ผ2 be as it is stated in the theorem. From (1/๐‘Ÿ)๐‘†๐œ†๐‘ฅ=๐‘ฆ we get to the system of equations: ๐‘๐‘ฅ0+๐‘ฅ1=๐‘ฆ0๐‘ฅ0+๐‘๐‘ฅ1+๐‘ฅ2=๐‘ฆ1๐‘ฅ1+๐‘๐‘ฅ2+๐‘ฅ3=๐‘ฆ2โ‹ฎ.(2.9) This is a nonhomogenous linear recurrence relation. Using the fact that (๐‘ฅ๐‘›),(๐‘ฆ๐‘›)โˆˆ๐‘0, for (2.9) we can reach a solution with generating functions. This solution can be given by ๐‘ฅ๐‘›=1๐›ผ21โˆ’1โˆž๎“๐‘˜=0๐‘ก๐‘›๐‘˜๐‘ฆ๐‘˜,(2.10) where ๐‘ก๐‘›๐‘˜=๎ƒฏ๐›ผ1๐‘›+1โˆ’๐‘˜โˆ’๐›ผ1๐‘›+3+๐‘˜๐›ผif๐‘›โ‰ฅ๐‘˜1๐‘˜+1โˆ’๐‘›โˆ’๐›ผ1๐‘˜+3+๐‘›if๐‘›<๐‘˜.(2.11) Let ๐‘‡=(๐‘ก๐‘›๐‘˜). We can see that by using Theorem 1.1, ๐‘‡โˆˆ๐ต(๐‘0). So (1/๐›ผ21โˆ’1)๐‘‡ is the resolvent operator of (1/๐‘Ÿ)๐‘†๐œ† and is continuous.

If ๐‘‡โˆถ๐œ‡โ†’๐œ‡ (๐œ‡ is โ„“1 or ๐‘0) is a bounded linear operator represented by the matrix ๐ด, then it is known that the adjoint operator ๐‘‡โˆ—โˆถ๐œ‡โˆ—โ†’๐œ‡โˆ— is defined by the transpose ๐ด๐‘ก of the matrix ๐ด. It should be noted that the dual space ๐‘โˆ—0 of ๐‘0 is isometrically isomorphic to the Banach space โ„“1 and the dual space โ„“โˆ—1, of โ„“1 is isometrically isomorphic to the Banach space โ„“โˆž.

Corollary 2.4. ๐œŽ(๐‘†,๐œ‡)โŠ‚[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ] for ๐œ‡โˆˆ{โ„“1,๐‘0,๐‘,โ„“โˆž}.

Proof. ๐œŽ(๐‘†,๐‘0)=๐œŽ(๐‘†โˆ—,๐‘โˆ—0)=๐œŽ(๐‘†,โ„“1)=๐œŽ(๐‘†โˆ—,โ„“โˆ—1)=๐œŽ(๐‘†,โ„“โˆž). And by Cartlidge [14], if a matrix operator ๐ด is bounded on ๐‘, then ๐œŽ(๐ด,๐‘)=๐œŽ(๐ด,โ„“โˆž). Hence we have ๐œŽ(๐‘†,๐‘0)=๐œŽ(๐‘†,โ„“1)=๐œŽ(๐‘†,โ„“โˆž)=๐œŽ(๐‘†,๐‘). What remains is to show that ๐œŽ(๐‘†,๐‘0)โŠ‚[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. By Theorem 2.3, there exists a resolvent operator of ๐‘†๐œ† which is continuous and the whole space ๐‘0 is the domain if the roots of the polynomial ๐‘ƒ(๐‘ฅ)=๐‘ฅ2+๐‘๐‘ฅ+1 satisfy ||๐›ผ2||||๐›ผ>1>1||.(2.12) So, if ๐œ†โˆˆ๐œŽ(๐‘†,๐‘0) then (2.12) is not satisfied. Since ๐›ผ1๐›ผ2=1, (2.12) is not satisfied means, the roots can be only of the form ๐›ผ1=1๐›ผ2=๐‘’๐‘–๐œƒ(2.13) for some ๐œƒโˆˆ[0,2๐œ‹). Then (๐‘žโˆ’๐œ†)/๐‘Ÿ=๐‘=โˆ’(๐›ผ1+๐›ผ2)=โˆ’(๐‘’๐‘–๐œƒ+๐‘’โˆ’๐‘–๐œƒ)=โˆ’2cos๐œƒ. Hence ๐œ†=๐‘ž+2๐‘Ÿcos๐œƒ, which means ๐œ† can be only on the line segment [๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ].

Theorem 2.5. ๐œŽ(๐‘†,๐œ‡)=[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ] for ๐œ‡โˆˆ{โ„“1,๐‘0,๐‘,โ„“โˆž}.

Proof. By Theorem 2.2 and Corollary 2.4(๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ)โŠ‚๐œŽ(๐‘†,โ„“โˆž)โŠ‚[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. Since the spectrum of a bounded linear operator over a complex Banach space is closed, we have ๐œŽ(๐‘†,โ„“โˆž)=[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. And from the proof of Corollary 2.4 we have ๐œŽ(๐‘†,โ„“1)=๐œŽ(๐‘†,๐‘0)=๐œŽ(๐‘†,๐‘)=๐œŽ(๐‘†,โ„“โˆž).

3. The Continuous Spectra and Residual Spectra

Lemma 3.1 (see [15, pageโ€‰59]). ๐‘‡ has a dense range if and only if ๐‘‡โˆ— is one to one.

Corollary 3.2. If ๐‘‡โˆˆ(๐œ‡โˆถ๐œ‡) then ๐œŽ๐‘Ÿ(๐‘‡,๐œ‡)=๐œŽ๐‘(๐‘‡โˆ—,๐œ‡โˆ—)โงต๐œŽ๐‘(๐‘‡,๐œ‡).

Theorem 3.3. ๐œŽ๐‘Ÿ(๐‘†,๐‘0)=โˆ….

Proof. ๐œŽ๐‘(๐‘†,โ„“1)=โˆ… by Theorem 2.1. Now using Corollary 3.2, we have ๐œŽ๐‘Ÿ(๐‘†,๐‘0)=๐œŽ๐‘(๐‘†โˆ—,๐‘โˆ—0)โงต๐œŽ๐‘(๐‘†,๐‘0)=๐œŽ๐‘(๐‘†,โ„“1)โงต๐œŽ๐‘(๐‘†,๐‘0)=โˆ….

Theorem 3.4. ๐œŽ๐‘Ÿ(๐‘†,โ„“1)=(๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ).

Proof. Similarly as in the proof of the previous theorem, we have ๐œŽ๐‘Ÿ(๐‘†,โ„“1)=๐œŽ๐‘(๐‘†โˆ—,โ„“โˆ—1)โงต๐œŽ๐‘(๐‘†,โ„“1)=๐œŽ๐‘(๐‘†,โ„“โˆž)โงต๐œŽ๐‘(๐‘†,โ„“1)=(๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ).

If ๐‘‡โˆถ๐‘โ†’๐‘ is a bounded matrix operator represented by the matrix ๐ด, then ๐‘‡โˆ—โˆถ๐‘โˆ—โ†’๐‘โˆ— acting on โ„‚โŠ•โ„“1 has a matrix representation of the form ๎‚ธ๐œ’0๐‘๐ด๐‘ก๎‚น,(3.1) where ๐œ’ is the limit of the sequence of row sums of ๐ด minus the sum of the limits of the columns of ๐ด, and ๐‘ is the column vector whose ๐‘˜th entry is the limit of the ๐‘˜th column of ๐ด for each ๐‘˜โˆˆโ„•. For ๐‘†๐œ†โˆถ๐‘โ†’๐‘, the matrix ๐‘†โˆ—๐œ† is of the form ๎‚ธ2๐‘Ÿ+๐‘žโˆ’๐œ†00๐‘†๐œ†๎‚น.(3.2)

Theorem 3.5. ๐œŽ๐‘Ÿ(๐‘†,๐‘)={๐‘ž+2๐‘Ÿ}.

Proof. Let ๐‘ฅ=(๐‘ฅ0,๐‘ฅ1,โ€ฆ)โˆˆโ„‚โŠ•โ„“1 be an eigenvector of ๐‘†โˆ— corresponding to the eigenvalue ๐œ†. Then we have (2๐‘Ÿ+๐‘ž)๐‘ฅ0=๐œ†๐‘ฅ0 and ๐‘†๐‘ฅโ€ฒ=๐œ†๐‘ฅโ€ฒ where ๐‘ฅโ€ฒ=(๐‘ฅ1,๐‘ฅ2,โ€ฆ). By Theorem 2.1,๐‘ฅโ€ฒ=(0,0,โ€ฆ). Then ๐‘ฅ0โ‰ 0. And ๐œ†=2๐‘Ÿ+๐‘ž is the only value that satisfies (2๐‘Ÿ+๐‘ž)๐‘ฅ0=๐œ†๐‘ฅ0. Hence ๐œŽ๐‘(๐‘†โˆ—,๐‘โˆ—)={2๐‘Ÿ+๐‘ž}. Then ๐œŽ๐‘Ÿ(๐‘†,๐‘)=๐œŽ๐‘(๐‘†โˆ—,๐‘โˆ—)โงต๐œŽ๐‘(๐‘†,๐‘)={2๐‘Ÿ+๐‘ž}.

Now, since the spectrum ๐œŽ is the disjoint union of ๐œŽ๐‘, ๐œŽ๐‘Ÿ, and ๐œŽ๐‘, we can find ๐œŽ๐‘ over the spaces โ„“1, ๐‘0, and ๐‘. So we have the following.

Theorem 3.6. For the operator ๐‘†, one has the following: ๐œŽ๐‘๎€ท๐‘†,โ„“1๎€ธ=๐œŽ{๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ},๐‘๎€ท๐‘†,๐‘0๎€ธ=[],๐œŽ๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ๐‘[(๐‘†,๐‘)=๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ).(3.3)

4. The Resolvent Operator

The following theorem is a generalization of Theorem 2.3.

Theorem 4.1. Let ๐œ‡โˆˆ{๐‘0,๐‘,โ„“1,โ„“โˆž}. The resolvent operator ๐‘†โˆ’1 over ๐œ‡ exists and is continuous, and the domain of ๐‘†โˆ’1 is the whole space ๐œ‡ if and only if 0โˆ‰[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. In this case, ๐‘†โˆ’1 has a matrix representation (๐‘ ๐‘›๐‘˜) defined by ๐‘ ๐‘›๐‘˜=1๐‘Ÿ๎€ท๐›ผ21๎€ธโ‹…๎ƒฏ๐›ผโˆ’11๐‘›โˆ’๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3๐›ผif๐‘›โ‰ฅ๐‘˜1โˆ’๐‘›+๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3if๐‘›<๐‘˜,(4.1) where ๐›ผ1 is the root of the polynomial ๐‘ƒ(๐‘ฅ)=๐‘Ÿ๐‘ฅ2+๐‘ž๐‘ฅ+๐‘Ÿ with |๐›ผ1|<1.

Proof. Let ๐œ‡ be one of the sequence spaces in {๐‘0,๐‘,โ„“1,โ„“โˆž}. Suppose ๐‘† has a continuous resolvent operator where the domain of the resolvent operator is the whole space ๐œ‡. Then ๐œ†=0 is not in ๐œŽ(๐‘†,๐œ‡)=[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. Conversely if 0โˆ‰[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ], then ๐‘† has a continuous resolvent operator, and since ๐‘† is bounded by Lemmaโ€‰7.2-7.3 of [12] the domain of this resolvent operator is the whole space ๐œ‡.
Now, suppose 0โˆ‰[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. Let ๐›ผ1 and ๐›ผ2 be the roots of the polynomial ๐‘ƒ(๐‘ฅ)=๐‘Ÿ๐‘ฅ2+๐‘ž๐‘ฅ+๐‘Ÿ where |๐›ผ1|โ‰ค|๐›ผ2|. Since 0โˆ‰[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ], by the proof of Corollary 2.4|๐›ผ1|โ‰ |๐›ผ2|. Then |๐›ผ1|<1<|๐›ผ2|. So ๐‘† satisfies the conditions of Theorem 2.3. Hence the resolvent operator of ๐‘† is represented by the matrix ๐‘†โˆ’1=(๐‘ ๐‘›๐‘˜) defined by ๐‘ ๐‘›๐‘˜=1๐‘Ÿ๎€ท๐›ผ21๎€ธโ‹…๎ƒฏ๐›ผโˆ’11๐‘›โˆ’๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3๐›ผif๐‘›โ‰ฅ๐‘˜1โˆ’๐‘›+๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3if๐‘›<๐‘˜,(4.2) when ๐œ‡=๐‘0 by that theorem. The matrix ๐‘†โˆ’1 is already a left inverse of the matrix ๐‘†. Observe that ๐‘†โˆ’1 satisfies also the corresponding conditions of Theorem 1.1, which means ๐‘†โˆ’1โˆˆ(๐œ‡,๐œ‡) for ๐œ‡โˆˆ{๐‘,โ„“1,โ„“โˆž}. So, the matrix ๐‘†โˆ’1 is the representation of the resolvent operator also for the spaces in {๐‘,โ„“1,โ„“โˆž}.

Remark 4.2. If a matrix ๐ด is a triangle, we can see that the resolvent (when it exists) is the unique lower triangular left hand inverse of ๐ด. In our case, ๐‘† is far away from being a triangle. The matrix ๐‘†โˆ’1 of this theorem is not the unique left inverse of the matrix ๐‘† for 0โˆ‰[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ]. For example, the matrix ๐‘‡=(๐‘ก๐‘›๐‘˜) defined by ๐‘ก๐‘›๐‘˜=1๐‘Ÿ๎€ท๐›ผ21๎€ธโ‹…๎ƒฏ๐›ผโˆ’11โˆ’๐‘›+๐‘˜+1โˆ’๐›ผ1๐‘›โˆ’๐‘˜+1if๐‘›<๐‘˜0if๐‘›โ‰ฅ๐‘˜(4.3) is another left inverse of ๐‘†. Then ๐œ†๐‘†โˆ’1+(1โˆ’๐œ†)๐‘‡ is also a left inverse of ๐‘† for any ๐œ†โˆˆโ„‚, which means there exist infinitely many left inverses for ๐‘†.

Theorem 4.3. Let 0โˆ‰[๐‘žโˆ’2๐‘Ÿ,๐‘ž+2๐‘Ÿ], and, ๐›ผ1 be the root of ๐‘ƒ(๐‘ฅ)=๐‘Ÿ๐‘ฅ2+๐‘ž๐‘ฅ+๐‘Ÿ with |๐›ผ1|<1. Then for ๐œ‡โˆˆ{๐‘0,๐‘,โ„“1,โ„“โˆž} we have โ€–โ€–๐‘†โˆ’1โ€–โ€–(๐œ‡โˆถ๐œ‡)=||๐›ผ1||+||๐›ผ1||2|||๐‘Ÿ|1โˆ’๐›ผ21||๎€ท||๐›ผ1โˆ’1||๎€ธ.(4.4)

Proof. Since ๐‘†โˆ’1 is a symmetric matrix, the supremum of the โ„“1 norms of the rows is equal to the supremum of the โ„“1 norms of the columns. So, according to Theorem 1.1, what we need is to calculate the supremum of the โ„“1 norms of the rows of ๐‘†โˆ’1. Denote the ๐‘›th row ๐‘†โˆ’1 by ๐‘†๐‘›โˆ’1 for ๐‘›=0,1,โ€ฆ. Now, let us fix the row ๐‘› and calculate the โ„“1 norm for this row. Let ๐œŒ=1/|๐‘Ÿ(1โˆ’๐›ผ21)|. By using Theorem 4.1, we have โ€–โ€–๐‘†๐‘›โˆ’1โ€–โ€–โ„“1๎ƒฉ=๐œŒ๐‘›๎“๐‘˜=0||๐›ผ1๐‘›โˆ’๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3||+โˆž๎“๐‘˜=๐‘›+1||๐›ผ1โˆ’๐‘›+๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3||๎ƒช๎ƒฉโ‰ค๐œŒ๐‘›๎“๐‘˜=0๎‚€||๐›ผ1||๐‘›โˆ’๐‘˜+1+||๐›ผ1||๐‘›+๐‘˜+3๎‚+โˆž๎“๐‘˜=๐‘›+1๎‚€||๐›ผ1||โˆ’๐‘›+๐‘˜+1+||๐›ผ1||๐‘›+๐‘˜+3๎‚๎ƒช๎ƒฉ=๐œŒ๐‘›๎“๐‘˜=0||๐›ผ1||๐‘›โˆ’๐‘˜+1+โˆž๎“๐‘˜=๐‘›+1||๐›ผ1||โˆ’๐‘›+๐‘˜+1+โˆž๎“๐‘˜=0||๐›ผ1||๐‘›+๐‘˜+3๎ƒช๎ƒฉ=๐œŒ๐‘›+1๎“๐‘˜=1||๐›ผ1||๐‘˜+โˆž๎“๐‘˜=2||๐›ผ1||๐‘˜+โˆž๎“๐‘˜=๐‘›+3||๐›ผ1||๐‘˜๎ƒช๎ƒฉ2=๐œŒโˆž๎“๐‘˜=0||๐›ผ1||๐‘˜||๐›ผโˆ’2โˆ’1||โˆ’||๐›ผ1||๐‘›+2๎ƒช๎ƒฉ2โ‰ค๐œŒโˆž๎“๐‘˜=0||๐›ผ1||๐‘˜||๐›ผโˆ’2โˆ’1||๎ƒช=||๐›ผ1||+||๐›ผ1||2|||๐‘Ÿ|1โˆ’๐›ผ21||๎€ท||๐›ผ1โˆ’1||๎€ธ.(4.5) Hence โ€–โ€–๐‘†โˆ’1โ€–โ€–(๐œ‡โˆถ๐œ‡)=sup๐‘›โ€–โ€–๐‘†๐‘›โˆ’1โ€–โ€–โ„“1โ‰ค||๐›ผ1||+||๐›ผ1||2|||๐‘Ÿ|1โˆ’๐›ผ21||๎€ท||๐›ผ1โˆ’1||๎€ธ.(4.6) On the other hand โ€–โ€–๐‘†๐‘›โˆ’1โ€–โ€–โ„“1๎ƒฉ=๐œŒ๐‘›๎“๐‘˜=0||๐›ผ1๐‘›โˆ’๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3||+โˆž๎“๐‘˜=๐‘›+1||๐›ผ1โˆ’๐‘›+๐‘˜+1โˆ’๐›ผ1๐‘›+๐‘˜+3||๎ƒช๎ƒฉโ‰ฅ๐œŒ๐‘›๎“๐‘˜=0๎‚€||๐›ผ1||๐‘›โˆ’๐‘˜+1โˆ’||๐›ผ1||๐‘›+๐‘˜+3๎‚+โˆž๎“๐‘˜=๐‘›+1๎‚€||๐›ผ1||โˆ’๐‘›+๐‘˜+1โˆ’||๐›ผ1||๐‘›+๐‘˜+3๎‚๎ƒช๎ƒฉ=๐œŒ๐‘›๎“๐‘˜=0|๐›ผ1|๐‘›โˆ’๐‘˜+1+โˆž๎“๐‘˜=๐‘›+1||๐›ผ1||โˆ’๐‘›+๐‘˜+1โˆ’โˆž๎“๐‘˜=0||๐›ผ1||๐‘›+๐‘˜+3๎ƒช๎ƒฉ=๐œŒ๐‘›+1๎“๐‘˜=1||๐›ผ1||๐‘˜+โˆž๎“๐‘˜=2||๐›ผ1||๐‘˜โˆ’โˆž๎“๐‘˜=๐‘›+3||๐›ผ1||๐‘˜๎ƒช.(4.7) Then โ€–โ€–๐‘†โˆ’1โ€–โ€–(๐œ‡โˆถ๐œ‡)=sup๐‘›โ€–โ€–๐‘†๐‘›โˆ’1โ€–โ€–โ„“1โ‰ฅlim๐‘›โ€–โ€–๐‘†๐‘›โˆ’1โ€–โ€–โ„“1๎ƒฉ=๐œŒโˆž๎“๐‘˜=1||๐›ผ1||๐‘˜+โˆž๎“๐‘˜=2||๐›ผ1||๐‘˜๎ƒช=||๐›ผ1||+||๐›ผ1||2|||๐‘Ÿ|1โˆ’๐›ผ21||๎€ท||๐›ผ1โˆ’1||๎€ธ.(4.8)


The author thanks the referees for their careful reading of the original paper and for their valuable comments.


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