Core invertibility of operators from the algebra generated by two orthogonal projections

A Hilbert space operator A is said to be core invertible if it has an inner inverse whose range coincides with the range of A and whose null space coincides with the null space of the adjoint of A. This notion was introduced by Baksalary, Trenkler, Rakić, Dinčić, and Djordjević in the last decade, who also proved that core invertibility is equivalent to group invertibility and that the core and group inverses coincide if and only if A is a so-called EP operator. The present paper contains criteria for core invertibility and for the EP property as well as formulas for the core inverse for operators in the von Neumann algebra generated by two orthogonal projections.


Introduction
Let H be a Hilbert space and B(H) be the algebra of all bounded linear operators acting on H. For any A ∈ B(H) we will denote by N (A) and R(A) the kernel and the range of A, respectively.
Recall that B ∈ B(H) is an inner inverse of A if ABA = A. The latter equality implies that AB is a projection onto R(A), and so A can only be inner invertible if its range is closed. As it happens, this condition is also sufficient and, if it holds, infinitely many inner inverses of A exist (unless, of course, A is invertible, in which case A −1 is the unique inner inverse of A). The additional requirement that A be also an inner inverse of B and that the projections AB, BA be orthogonal single out a unique inner inverse, the so-called Moore-Penrose inverse of A, usually denoted by A † .
The second named author was supported in part by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi.
It was proved in [10] that A # exists only simultaneously with A # and that the two coincide exactly when A # = A † , i.e., exactly for EP operators A (Theorems 3.2 and 3.10 of [10], respectively).
In this paper, we take a closer look at the operators from the von Neumann algebra W * (P, Q) generated by two orthogonal projections P, Q in B(H). A constructive criterion for Moore-Penrose invertibility of operators A ∈ W * (P, Q) goes back to [11], see also [4,Theorem 8.1]. Drazin inversion was dealt with in [3] and [4,Sect. 9]. Since the index of Drazin invertible operators A ∈ W * (P, Q) was also computed there, implicitly the group invertibility criterion and formulas for A # were established there as well. For convenience of reference we state them explicitly in Sect. 2, along with the required notation. The rest of the section is devoted to the core invertibility. The EP property of operators from W * (P, Q) is studied in Sect. 3. Some examples are examined in Sect. 4.
To conclude this introduction, let us mention our paper [5], in which the previously known results on the operators in W * (P, Q) mentioned above (and some more) are stated and discussed from a unifying point of view. The main idea we were trying to convey in [5] was that any reasonable question about operators from the algebra generated by P and Q can be answered based on the structural description of the (P, Q) pair going back to the "two subspaces" theory by Halmos [9]. This paper is yet another illustration of this point.
Core invertibility of operators from the algebra 259

The core inverse
Our main tool is the explicit formula for operators in W * (P, Q) established in [8]; see also [4,Sect. 7]. To introduce the pertinent language, recall first of all Halmos' "two subspaces" approach according to which H is represented as the orthogonal sum of The compression H of P (I − Q)P onto M 0 and that of (I − P )Q(I − P ) onto M 1 are then unitarily similar. Identifying M 1 with M 0 via this unitary similarity, operators A ∈ W * (P, Q) take the form Here α ij ∈ C (i, j = 0, 1), (α 00 , α 01 , α 10 , α 11 ) is an abbreviated notation for α 00 I M00 ⊕ α 01 I M01 ⊕ α 10 I M10 ⊕ α 11 I M11 , and the entries of the matrix . Then (2) can be refined to The Drazin invertibility criterion and the formulas for both A D and the index of A from [3,4] yield the following.

Theorem 1.
Let A ∈ W * (P, Q) be represented in the form (5). Then for A # to exist it is necessary and sufficient that det Φ A and Tr Φ A are separated from zero on Δ 2 (A) and Δ 1 (A), respectively. Under these conditions Here and below for any z ∈ C we let z † = z −1 if z = 0 while 0 † = 0. This notation is justified by the simple fact that z † is indeed the Moore-Penrose inverse of z in the (trivial) one-dimensional case. We also recall that a function f is said to be separated from zero on some set E if the infimum of |f | on E is strictly positive. Before stating the respective result for the core inverse, let us introduce one more notation: Theorem 2. Let A ∈ W * (P, Q) be represented in the form (5). Then for A # to exist it is necessary and sufficient that det Φ A and Tr Φ A are separated from zero on Δ 2 (A) and Δ 1 (A), respectively. Under these conditions where Proof. The existence conditions are the same as in Theorem 1 due to [10, Theorem 3.8]. When they hold, the right hand side of (7) is defined correctly.
Checking that it matches the definition of A # can be done block-wise, and the only not quite trivial part is that and as both sides are invariant under unitary similarity, we may assume that Φ A is lower triangular. Due to the rank one condition, we may even assume that Φ A = u 0 v 0 with |u| 2 + |v| 2 > 0. This reduces the verification of the equality to a simple computation. Furthermore, the operator (φ A Tr Φ A )(H 1 ) is invertible and commutes with the blocks of Φ A (H 1 ) and Φ * A (H 1 ). From (8) it therefore follows that

Corollary 3. Let A ∈ W * (P, Q) be group-(and thus core-) invertible. Then
Proof. From (6) and (7) with the use of (8) it is clear that (9) holds if and only if So, we need to check when a rank one 2-by-2 matrix X is such that If X is normal, then X = U diag(λ, 0)U * with a unitary matrix U and a nonzero number λ, and since the trace and the Frobenius norm are unitarily equivalent, equality (10) amounts to the identity λ = (λ/|λ| 2 )|λ| 2 . Conversely, suppose (10) holds. As the right-hand side is a scalar multiple of a Hermitian matrix, it follows that so also is the X on the left, which implies that X is normal.
Core invertibility of operators from the algebra 261 (Incidentally, an alternative argument can be based on the idea used in the previous proof. Namely, since all the ingredients in (10) are unitarily invariant, we may without loss of generality suppose that X = u 0 v 0 with |u| 2 + |v| 2 > 0. Then (10) takes the form Obviously, this holds if and only if v = 0.) Along with the core inverse, the authors of [10] have also introduced its dual. Namely, an inner inverse of the operator A is called its dual core inverse, denoted by A # , if in place of (1) It is easy to check that the analogue of Theorem 2 holds almost literally; one just needs to replace A # with A # in (7) and switch the positions of Φ A and Φ * A in (8). Consequently, Corollary 3 with A # replaced by A # holds as well.

EP property
When combined with [10, Theorem 3.10] mentioned in the Introduction, Corollary 3 implies that a group invertible operator A ∈ W * (P, Q) possesses the EP property if and only if Φ A is normal on Δ 1 (A). However, we would like to treat the EP property of operators in W * (P, Q) without any additional a priori conditions. The main step in this direction is to figure out what the normality of Φ A on Δ 1 (A) is equivalent to in this setting.

Proposition 4. Let A ∈ W * (P, Q). Then N (A) = N (A * ) if and only if the matrix-function Φ A from its representation (2) is normal on Δ 1 (A).
Proof. We will use the kernel description of A in terms of its representation (5). Recall therefore that M (1) is the spectral subspace of A corresponding to the Δ 1 (A) part of its spectrum and that H 1 is the compression of P (I − Q)P to M (1) . According to [  Here and θ = η/ |η| with η = φ 00 φ 01 + φ 10 φ 11 (13) (as in [11], θ(t) is unimodular with an arbitrarily chosen argument for values of t where η(t) = 0).
From the proof of Proposition 4 it can be seen that for operators A ∈ W * (P, Q) the kernels of A and A * are nested (i.e., one is contained in the other) only when they coincide.
Recall now that for any X ∈ B(H) the closure of R(X) is nothing but N (X * ) ⊥ . So, X is an EP operator if and only if its range is closed and N (X * ) = N (X). A range-closedness criterion for operators in W * (P, Q) is known [11,Sect. 2] (see also [4,Theorem 7.7]) and reads as follows. Combining Propositions 4 and 5, we arrive at the following. Theorem 6. Let A ∈ W * (P, Q). Then A is an EP operator if and only if the determinant of Φ A is separated from zero on Δ 2 (A) and Φ A is a normal matrix on Δ 1 (A) whose Frobenius norm is separated from zero on Δ 1 (A).
Let P and Q be the orthogonal projections onto R(T ) and N (T ), respectively, and represent these two operators in the form (3) and (4). It is well known from [1] (also cited as Proposition 1.6 in [4]) that then P Q < 1 and that T may be written as T = (I − P Q) −1 P (I − P Q). Thus, T and hence also A are in W * (P, Q).
It follows in particular that T + αT * (α = 0) is core invertible if and only if 1 / ∈ σ(H) and that the so-called Buckholtz operator T + T * − I is always core invertible. Here are two concrete examples of the cases (i) and (ii) of Theorem 7.