Subnormal operators whose adjoints have rich point spectrum

Abstract

A generalized version of the Glauber-Klauder basic formula of quantum optics is shown to be valid for any cyclic subnormal operator S whose adjoint has a rich point spectrum σ p (S * ) (in the sense that a semispectral measure of S vanishes on C \ σ p (S * ) * ). It is exhibited that such operators always have analytic models. The point spectrum of the adjoint of a subnormal operator which satisfies a generalized version of the Glauber-Klauder formula is proved to be rich (in the above sense).

Analytic structure of weighted shifts on directed trees

“…Hence, applying linearity, we see that the equality in (13) holds with every f ∈ E. This, continuity of V w and density of E in ℓ 2 (β) implies (13).…”

Borel structure of the spectrum of a closed operator

“…In particular, the above theorem applies to the point spectrum of a closed operator in H, which solves the problem posed in [7] (see Question after Example 2 there). Partial solution of this issue for certain classes of closed operators in Hilbert spaces may be found e.g.…”

“…Let H be a separable Hilbert space. As it is easily seen, any subset of the complex plane is the point spectrum of an unbounded linear operator acting on H. To see this, note that the dimension of H as a vector space is equal to the power of the continuum; take a Hamel basis {e s } s∈R of H and any function ψ : R → C, define A : H → H by the rule Ae s = ψ(s)e s , and observe that the point spectrum of A coincides with the image of the function f (see also [7,Example 2]). So, non-Borel subsets of C may be the point spectra of certain linear operators acting on H. It is also worth while to mention that every bounded subset of C is the point spectrum of a bounded normal (even diagonal) operator on a nonseparable Hilbert space.…”

Composition operators on Hilbert spaces of entire functions with analytic symbols

“…This fact was proved in [23,24] (see also [25]) under the additional assumption that ν({0}) = 0. However, arguing as in [36,Example 18] (with emphases put on [36, Eq. ( 40)]), one can show that (3.11) remains true without assuming that ν({0}) = 0.…”