4 Operators on Partial Inner Product Spaces: Towardsa Spectral Analysis (original) (raw)

Operators on Partial Inner Product Spaces: Towards a Spectral Analysis

Mediterranean Journal of Mathematics, 2014

Given a LHS (Lattice of Hilbert spaces) VJ and a symmetric operator A in VJ , in the sense of partial inner product spaces, we define a generalized resolvent for A and study the corresponding spectral properties. In particular, we examine, with help of the KLMN theorem, the question of generalized eigenvalues associated to points of the continuous (Hilbertian) spectrum. We give some examples, including so-called frame multipliers.

Spectral theory of linear operators

Advances in Mathematics, 1983

This thesis is concerned with the relationship between spectral decomposition of operators, the functional calculi that operators admit, and Banach space structure.

Generalized frames for operators in Hilbert spaces

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2014

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ-g-frame, where Θ is a bounded operator on a Hilbert space. Θ-g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ-g-frames by considering Θ-g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

Partial inner product spaces, metric operators and generalized hermiticity

Journal of Physics A: Mathematical and Theoretical, 2012

Motivated by the recent developments of pseudo-hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (pip-space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular pip-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism.

Resolvents of functions of operators with Hilbert-Schmidt hermitian components

Filomat, 2018

Let H be a separable Hilbert space with the unit operator I. We derive a sharp norm estimate for the operator function (?I-f(A))-1 (? ? C), where A is a bounded linear operator in H whose Hermitian component (A- A*)/2i is a Hilbert-Schmidt operator and f(z) is a function holomorphic on the convex hull of the spectrum of A. Here A* is the operator adjoint to A. Applications of the obtained estimate to perturbations of operator equations, whose coefficients are operator functions and localization of spectra are also discussed.

A Spectral Characterization of Operators

Journal of the Australian Mathematical Society, 2016

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.