Spectral Properties of a Multiplication Operator

Abstract

Abstract. In this note we study the spectral properties of a multiplication operator in the space Lp(X)" which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called "floating singularities" .

Spectra of Multiplication Operators in Sobolev Spaces

“…The following is often stated without proof, see e. g. [6]; therefore we shall include a proof for the sake of completeness. Proof.…”

“…A comprehensive description of spectral properties of such multiplication operators in Lebesgue spaces can be found in [6]. Indeed, for a measure space (X, Σ, μ) consider a d×d matrix function whose entries are measurable functions on X.…”

“…Properties of the spectrum of A have been discussed in [6]. Multiplication-like operators also occur with direct integrals, see [4,5].…”

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

“…From the proof of Proposition 4.9 we already know that λ ∈ σ ess (L) if and only if Z r1 11 −λ and M 4 −λ are Fredholm operators on L 2 (0, r 1 ), 1 r 3 and L 2 (r 1 , r 0 ), 1 r 2 , respectively, for any r 1 ∈ (0, r 0 ]. But M 4 − λ is invertible by [7,Proposition 2.3]. Note that Z r1 11 has the same form as Z, just considered on a subinterval, because in the definition of V + f we can replace the upper limit r 0 in the integral by r 1 since the function f has support in [0, r 1 ].…”

The Spectrum of the Multiplication Operator Associated with a Family of Operators in a Banach Space

“…For not necessarily self-adjoint operators in Hilbert space one can use the theory of direct integrals, see e. g. Azoff, [1], and Dixmier, [5,Chapter II,§2]. For usual multiplication operators, i. e. multiplication by matrix functions, the spectrum has been investigated e. g. by Hardt and Wagenführer in [7].…”