Abstract
Let be a complex separable Hilbert space and be the algebra of all bounded linear operators from to . Our goal in this article is to describe the closure of numerical range of parallel sum operator for two orthogonal projections and in as a closed convex hull of some explicit ellipses parameterized by points in the spectrum.
1. Introduction
Let be a complex separable Hilbert space with inner product and be the algebra of bounded linear operators on . The numerical range of an operator is defined as
It is known that is a nonempty bounded convex set in the complex plane and its closure, denoted by , always contains the spectrum of T (see [1, 2]). In addition, for , we have , where stands for the convex hull of the set . For references on the numerical range and its generalizations, see, for instance, [3–8].
This paper arose from an attempt to gain a geometric characterization of the numerical range of parallel sum with a view of operator block. In what follows we always suppose and has closed range. The parallel sum of and is defined aswhere is the Moore–Penrose generalized inverse of T (see [9, 10]). The study of parallel sum is motivated by the fact that if A and B are impedance operators of resistive -port electrical networks, then is the impedance operator of the parallel connection [11]. Several authors, in particular Anderson and Trapp [11], Anderson and Duffin [12], Ando [13], and Wang et al. [10], extended this result and established many different equivalent definitions and properties on parallel sum (see also [9, 10]). Recently, Klaja [14] applied Halmos’ two projections theorem to describe the numerical range of a product of two orthogonal projections and . He showed that the closure of its numerical range is equal to a closed convex hull of some ellipses parametrized by points in the spectrum. In [8], Wang et al. also used Halmos’ two projections theorem to study the containment region of the numerical range of the product of a pair of positive contractions. Zhang and Yu [15] described the numerical range of the operator . Motivated by these, we consider the numerical range of the parallel sum for orthogonal projections and . The investigation uses in an essential way Halmos’ two projections theorem, which is introduced as follows.
Let and be two orthogonal projections on . Thus, and . The ranges of and are denoted by and , respectively. According to Halmos’ two projections theorem (see [16] and consult [17] for the history and more on the subject), there is a representation of as an orthogonal sum:where , , . If one of the spaces and is nontrivial, then these two spaces have the same dimension and there exist two self-adjoint operators and of into itself such that , , , and such that and are simultaneously unitary equivalent to the following operator matrices:
Moreover, there exists a self-adjoint operator verifying such that and .
In [18], Deng and Du introduced the pair in generic position, if . If two orthogonal projections and are in generic position, then and the operator matrices in (4) can be simplified to
Tian et al. [9] gave a specific matrix representation of the operator with respect to the decomposition (1). Let and have the operator matrices in (4), and we can get
If and are in generic position, the above operator matrix in turn will bewhich will be very useful in the next section.
2. Main Results
The following theorems are the main results of this article. Let , . We denote the domain delimited by the ellipse with foci 0 and , and minor axis length
Theorem 1. Let , be two orthogonal projections; then, for , the closure of the numerical range of operator is the closed convex hull of the elliptical disk :
If are in generic position, we will first prove the following theorem.
Theorem 2. Let , be two orthogonal projections in generic position; then, for , the closure of the numerical range of operator is the closed convex hull of the elliptical disk :
In order to prove Theorem 2, we need the following definition and lemmas.
Definition 3. (see [14]). Let be a bounded convex set in . Let . The support function of , of angle , is defined by the following formula:
Lemma 4 (see [14, 19]). We denote by the closure of . We have
Lemma 5 (see [14, 19]). Let be two bounded convex sets of the plane with support functions and , respectively. Let be such that . Then, we have
Lemma 6. Let be in generic position. Then, the support function of the numerical range of operator is
Proof. We fix . From Definition 3, we can getIt follows thatand we haveFromwhere , it follows that . After some computations, we can get withandwhereIt is easy to verify passing to the limit when goes to thatWe also have that . As all entries of are Borelians functions and is a self-adjoint operator, according to Borelians functional calculus (see [20]), we can defineThen, we also can define and , and we have thatSo, we obtainNote that for every and . Since , we also have that , and then we obtainFrom (5) and (7), we can getIt follows thatand we haveDenoting , where , then . We obtain thatThis completes the proof.
In order to describe clearly, we characterize it as the closed convex hull of ellipses . Several of these ellipses are shown in Figure 1.
Remark 7. The Cartesian equation of the boundary of is given byand the parametric equation of the boundary of is given bywhere .
Lemma 8. Let . The support function of the elliptical disk is
Proof. Let . The support function of relative to the original point 0 is given bywhere represent the parametric equation of the boundary of . Let be the function defined by the following formula:Since is symmetric about , only needs to be considered, and the proof will be divided into two cases.
Case One. Suppose that . It follows fromthat we have if and only if . So, the critical points are and . We denoted ; then,where . Substituting the above formula into , we can getBy simple calculation, we haveThen, we finally get thatCase Two. If , then . The function reaches its maximum value while and also satisfiesThe proof is completed.
Then, we can prove Theorems 1 and 2.
Proof of Theorem 1. From Lemmas 6 and 8, we haveIt follows from Lemma 6 thatThe proof is completed.
Proof of Theorem 2. From the matrix form in (2), we haveSuppose . The following proof will be divided into two cases.(1)If , then . In this case, and . Thus, .(2)If , we have .Suppose . The following proof will be divided into two cases.(3)If , we have on the space . Thus, the closure of the numerical range of on the space is . As for all , we can have on the space .(4)If , we have on the space . Thus, the closure of the numerical range of on the space is . As for all and , the convex combination of 0 and is contained in the closure of the numerical range of . Thus, the closure of the numerical range of on the space is . But . So, we have on the space . The proof is completed.
Corollary 9. Let and be orthogonal projections. Then, for , we can get
In particular, we have when , as shown in Figure 2 [21–24].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in this paper. All authors have read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 12061031 and 11461018) and the Natural Science Basic Research Plan in Hainan Province of China (nos. 120MS030 and 123RC473).