Let T be a contraction on a complex Hilbert space H , let( ) x T σ be the local spectrum of T at H x ∈ , and let lim , where u T x is the unitary part of x in the canonical decomposition of H with respect to T .
Let T = {T (t)} t∈R be a C 0 -group on a complex Banach space X dominated by a weight function ω(t) = (1 + |t|) α (0 α < 1) and let A be its generator with domain D( A). Among other things, it is shown that if the operator A has compact local spectrum at x ∈ X, then x ∈ D( A) and there exist double sequences of real numbers (c n ) n∈Z and (t n ) n∈Z such thatis the local spectral radius of A at x. As an application, some inequalities of Bernstein type in L p -spaces are given.
Let A be a complex commutative Banach algebra and let MA be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points φ1 , φ2 ∈ MA , there is an element a ∈ A for whichâ (φ1 ) = 1,â (φ2 ) = 0 and a ≤ C, whereâ is the Gelfand transform of a ∈ A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range.
In this study, we aim to present some new versions of classical Schauder and Banach fixed point theorems under weak topology in general Banach spaces. We have also another main goal which is to prove that Krasnoselskii theorem ensures in general Banach spaces using these new results. As an application, an illustrative example providing such results is given. Our results extend known results on the issue.
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