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Complex Analysis and Operator Theory

Generalized Inverses of Conditional Type Operators H. Emamalipour1 · M. R. Jabbarzadeh1 · A. Shahi1 Received: 15 February 2020 / Accepted: 28 August 2020 © Springer Nature Switzerland AG 2020

Abstract In this paper, some characterizations of the Drazin and the Moore–Penrose inverses of the conditional type operators on L 2 () are established. Keywords Conditional expectation · Drazin inverse · Moore–penrose inverse · Finite-rank Mathematics Subject Classification Primary 47B20; Secondary 47B25

1 Introduction and Preliminaries Let H and K be separable complex Hilbert spaces with inner product  , . Let B(H, K) be the set of all bounded linear operators from H into K and let BC (H, K) be the subspace of all T ∈ B(H, K) such that the range of T is closed in K. If H = K, we write B(H) = B(H, K) and BC (H) = BC (H, H). For T ∈ B(H, K), N (T ) and R(T ) denote the kernel and the range of T , respectively. The Moore–Penrose inverse of T ∈ B(H, K) is the operator S ∈ B(K, H) which satisfies the Penrose equations (1) T ST = T , (2) ST S = S, (3) (T S)∗ = T S, (4) (ST )∗ = ST .

(1.1)

Communicated by Ilwoo Cho. This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory (Marek Bozejko, Palle Jorgensen and Yuri Kondratiev”.

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M. R. Jabbarzadeh [email protected] H. Emamalipour [email protected] A. Shahi [email protected]

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Faculty of Mathematical Sciences, University of Tabriz, 5166615648 Tabriz, Iran 0123456789().: V,-vol

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H. Emamalipour et al.

The Moore–Penrose inverse of T exists if and only if R(T ) is closed in K. If the Moore– Penrose inverse of T exists, then it is unique, and it is denoted by T † . Let T {i, . . . , j} denote the set of all operators S which satisfy the equations (1) ≤ (i), . . . , ( j) ≤ (4). In this case S ∈ T {i, . . . , j} is a {i, . . . , j}-inverse of T and is denoted by T (i,..., j) . Noth that T (1,2,3,4) = T † . An element T ∈ B(H) is said to have a Drazin inverse, or T is Drazin invertible if there exists S ∈ B(H) such that ST S = S, T S = ST and T k+1 S = T k for some k ∈ N. The minimal such k is called the Drazin index of T , and will be denoted by ind(T ). If T has Drazin inverse, then it is unique and denoted by T D . When k = 1, the Drazin inverse reduced to the group inverse and it is denoted by T # . Recall that asc(T ) and des(T ), the ascent and descent of T ∈ B(H), is the smallest non-negative integer n such that N (T n ) = N (T n+1 ) and R(T n ) = R(T n+1 ), respectively. It is well known that asc(T ) = des(T ) if asc(T ) and des(T ) are finite (see [16]). For T ∈ B(H), T D exists if and only if T has finite ascent and descent. In this case, ind(T ) = asc(T ) = des(T ) = n. For other important properties of T † and T D , see e.g. [1,3]. Let H = H1 ⊕ H2 , T ∈ B(H) and  let P j : H → H be an orthogonal T11 T12 , where Ti j : H j → Hi projection onto H j for j = 1, 2. Then T = T21 T22 is the operator given by Ti j = Pi T P j |H j . In particular, T (H1 ) ⊆