Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted KQA * , of operators satisfyingwhere k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point µ of the spectrum of T ∈ KQA * , then E is self-adjoint and EH = ker(T − µ) = ker (T − µ) * . Some spectral properties are also presented.
We introduce the class of [Formula: see text]-quasi-[Formula: see text]-isometric operators on Hilbert space. This generalizes the class of [Formula: see text]-isometric operators on Hilbert space introduced by Agler and Stankus. An operator [Formula: see text] is said to be [Formula: see text]-quasi-[Formula: see text]-isometric if [Formula: see text] In this paper [Formula: see text] matrix representation of a [Formula: see text]-quasi-[Formula: see text]-isometric operator is given. Using this representation we establish some basic properties of this class of operators.
The general problem in this paper is minimizing the C∞− norm of suitable affine mappings from B(H) to C∞, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality.
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