Abstract
We describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space ℋ; some related consequences are discussed. Furthermore, we show that two densely similar cyclic Banach‐space operators possessing Bishop′s property (β) have equal approximate point spectra.
Highlights
In the present paper, all Banach spaces are complex
For an operator T ∈ ᏸ(ᐄ), let T ∗, σ (T ), ρ(T ) := C\σ (T ), σp(T ), σap(T ), Γ (T ), ker T, and ran T denote the adjoint operator acting on the dual space ᐄ∗, the spectrum, the resolvent set, the point spectrum, the approximate point spectrum, the compression spectrum, the kernel, and the range, respectively, of T
For an operator T ∈ ᏸ(ᐄ), let (T ) denote the open set of complex numbers λ ∈ C for which there exists a nonzero analytic function φ : ᐂ → ᐄ on some open disc ᐂ centered at λ such that (T − μ)φ(μ) = 0 ∀μ ∈ ᐂ
Summary
All Banach spaces are complex. Let ᐄ be a Banach space and let ᏸ(ᐄ) denote the algebra of all linear bounded operators on ᐄ. We use this result and show that densely similar cyclic Banach-space operators possessing Bishop’s property (β) have the same approximate point spectra; this result generalizes [12, Theorem 4]. For every open subset U of C, we let ᏻ(U, ᐄ) denote the space of analytic ᐄ-valued functions defined on U .
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