Abstract
AbstractWe characterize skew-symmetric operators on a reproducing kernel Hilbert space in terms of their Berezin symbols. The solution of some operator equations with skew-symmetric operators is studied in terms of Berezin symbols. We also studied essentially unitary operators via Berezin symbols.
Highlights
We denote by a complex separable infinite dimensional Hilbert space endowed with the inner product 〈⋅,⋅〉, and ( ) the algebra of all bounded linear operators on
We study in terms of the Berezin symbols the solvability of some operator equations with skew-symmetric operators
We study the solution of the operator equations TX = K + Y and XT = L + in some Engliš C∗-operator algebras of operators on the reproducing kernel Hilbert spaces, including the Hardy space H2 = H2( ) over the unit disc of the complex plane
Summary
We denote by a complex separable infinite dimensional Hilbert space endowed with the inner product 〈⋅,⋅〉, and ( ) the algebra of all bounded linear operators on. For any self-adjoint operator T on the operator iT is skew-symmetric As it is known, many classical results in the matrix theory deal with complex symmetric matrices (i.e., T = Tt) and skew-symmetric matrices (i.e., T = −Tt). Li and Zhu [4] studied the skew-symmetry of normal operators and gave two structure theorems of skew-symmetric normal operators. We characterize skew-symmetric operators on an RKHS in terms of their Berezin symbols. We study in terms of the Berezin symbols the solvability of some operator equations with skew-symmetric operators. Note that such an approach, apparently, was initiated by the second author. We give in terms of the Berezin number a sufficient condition providing essential unitarity of essentially invertible operators
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