Universal Decomposition Equalities for Operator Matrices in a Hilbert Space
- PDF / 305,114 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 51 Downloads / 206 Views
Complex Analysis and Operator Theory
Universal Decomposition Equalities for Operator Matrices in a Hilbert Space Bo Jiang1 · Yongge Tian2 Received: 4 August 2020 / Accepted: 28 August 2020 © Springer Nature Switzerland AG 2020
Abstract This paper establishes some universal decomposition equalities for operator matrices in a Hilbert space. It includes two basic universal operator matrix decompositions for twoby-two and four-by-four operator matrices, and two four-by-four universal operator matrix decompositions for a four-term linear combination x0 I + x1 P + x2 Q + x3 P Q, where P and Q are two commutative involutory or two commutative idempotent operators, and x0 , x1 , x2 , x3 are four complex scalars. Many consequences are also presented concerning disjoint decomposition equalities, inverses, generalized inverses, collections of involutory, idempotent and tripotent operators generated from these linear combinations, etc. Keywords Hilbert space · Involutory operator · Idempotent operator · Linear combination · Decomposition Mathematics Subject Classification 47A05 · 47A08 · 47A68
1 Introduction Let H and K be two complex Hilbert spaces and let B(K, H) be the set of all bounded linear operators from K to H and abbreviate B(K, H) to B(H) if K = H. An A ∈ B(H) is said to be invertible if and only if there existsX ∈ B(H) such that AX = X A = I , the identity operator. An operator X ∈ B(H) is said to be a group inverse
Communicated by Daniel Aron Alpay.
B
Bo Jiang [email protected] Yongge Tian [email protected]
1
College of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, China
2
College of Business and Economics, Shanghai Business School, Shanghai, China 0123456789().: V,-vol
70
Page 2 of 15
B. Jiang, Y. Tian
of operator A ∈ B(H) if AX A = A, X AX = X , and AX = X A hold and is denoted by X = A# . An operator A ∈ B(H) is said to be involutory if A2 = I ; idempotent if A2 = A; tripotent if A3 = A. An idempotent operator is often called oblique projector whose null space is oblique to its range, in contrast to orthogonal projector, whose null space is orthogonal to its range. It has already been known that involutory, idempotent, and tripotent elements can be defined almost in all algebraic systems, and have been viewed as the building blocks of the systems for a long time. On the other hand, they are usually taken as useful study tools in the investigation of many problems in operator algebras due to their simple and nice operational properties, so that researches and contributions concerning these basic objects have long-lasting influence on many other topics in various algebraic systems (cf. [1–16,22,23]). As always, algebraists have proposed various pure and applied research problems in a Hilbert space framework, including various operator equalities, operator identities, and operator equations from theoretical and applied points of view, some of which have deep roots and counterparts in the complex matrix algebra. In an earlier paper [17], Tian
Data Loading...
