On Unitary Quasi-Equivalence of Operators

Georgian Mathematical Journal, 1996

Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented.

Springer Proceedings in Physics, 2016

Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties. Then we study quasi-Hermitian operators, bounded or not, that is, operators that are quasi-similar to their adjoint and we discuss their application in pseudo-Hermitian quantum mechanics. Finally, we extend the analysis to operators in a partial inner product space (pip-space), in particular the scale of Hilbert spaces generated by a single unbounded metric operator.

Comptes Rendus Mathematique, 2012

Given self-adjoint operators A, B ∈ B(H) it is said A ≤ u B whenever A ≤ U * BU for some unitary operator U. We show that A ≤ u B if and only if f (g(A) r) ≤ u f (g(B) r) for any increasing operator convex function f , any operator monotone function g and any positive number r. We present some sufficient conditions under which if B ≤ A ≤ U * BU , then B = A = U * BU. Finally we prove that if A n ≤ U * A n U for all n ∈ N, then A = U * AU. A ≤ u B ⇒ e A ≤ u e B. (1) Okayasu and Ueta [7] gave a sufficient condition for a triple of operators (A, B, U) with A, B ∈ B h (H) and U ∈ U(H) under which B ≤ A ≤ U * BU implies B = A = U * BU. In this note we use their idea and prove a similar result. In fact we present some sufficient conditions on an operator U ∈ U(H) for which B ≤ A ≤ U * BU ensures B = A = U * BU when A, B ∈ B h (H). It is known that ≤ u satisfies the reflexive and transitive laws but not the antisymmetric law in general; cf. [7]. The antisymmetric law states that A ≤ u B and B ≤ u A ⇒ A, B are unitarily equivalent. We, among other things, study some cases in which the antisymmetric law holds for the relation ≤ u. We refer the reader to [4] for general information on operators acting on

In this paper, we investigate almost-similarity relation, metric equivalence and other relations of operators on Hilbert spaces. We characterize almost-similar, metrically equivalent operators. Roughly speaking, we consider two operators to be " equivalent " if they are " close " to each other in some sense. We establish some spectral invariants with respect to each of these equivalence relations. B. M. NZIMBI et al. 2

Bulletin of the London Mathematical Society, 1979

Let T be in S£(ffl), the algebra of bounded linear operators on the complex Hilbert space Jif. Let W(T) be the weakly closed algebra generated by T and / , let Lat T be the lattice of closed invariant subspaces of T, and let Alg Lat T = {AE $e(2te): Lat T cz Lat A}. Then Tis reflexive if iV{T) -Alg Lat T. Some results on reflexive operators occur in [2], [3], [5], and [7]. In particular, Sarason ([7]) shows that normal operators and analytic Toeplitz operators are reflexive, and Deddens ([2]) shows that isometries are reflexive. Suppose that T x and T 2 are reflexive. It is not known ([3], [5])

Filomat, 2007

For Hilbert space operators A and B, let ?AB denote the generalized derivation ?AB(X) = AX - XB and let /\AB denote the elementary operator rAB(X) = AXB-X. If A is a pk-quasihyponormal operator, A ? pk - QH, and B*is an either p-hyponormal or injective dominant or injective pk - QH operator (resp., B*is an either p-hyponormal or dominant or pk - QH operator), then ?AB(X) = 0 =? SA*B*(X) = 0 (resp., rAB(X) = 0 =? rA*B*(X) = 0). .

Journal of Mathematical Physics, 2014

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

Linear Algebra and its Applications, 2005

Let H be an infinite-dimensional complex Hilbert space. We give the characterization of surjective mappings on B(H) that preserve unitary similarity in both directions.

2007

For Hilbert space operators A and B, let δ AB denote the generalised derivation δ AB (X) = AX − XB and let AB denote the elementary operator AB (X) = AXB−X. If A is a pk-quasihyponormal operator, A ∈ pk − QH, and B * is an either p-hyponormal or injective dominant or injective pk − QH operator (resp., B * is an either p-hyponormal or dominant or pk − QH operator), then δ AB (X) = 0 =⇒ δ A * B * (X) = 0 (resp., AB (X) = 0 =⇒ A * B * (X) = 0).

Bulletin of The Iranian Mathematical Society, 2011

Let L(B(H)) be the algebra of all linear operators on B(H) and P be a property on B(H). For 1, 2 2 L(B(H)), we say that 1 P 2, whenever 1(T) has property P, if and only if 2(T) has this property. In particular, if I is the identity map on B(H), then PI means that preserves property P in both directions. Each property P produces an equivalence relation on L(B(H)). We study the relation between equivalence classes with respect to dif- ferent properties such as being Fredholm, semi-Fredholm, compact, finite rank, generalized invertible, or having a specific semi-index. Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. We denote by F(H) and K(H) the ideals of all finite rank and compact operators in B(H), respectively. The Calkin algebra of H is the quotient algebra C(H) = B(H)/K(H). An operator T 2 B(H) is said to be a Fredholm operator if Im(T), the range of T, is closed and both its kernel and co-kernel are fin...