This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D706})$. This is a new criterion for determining the multiplicities of eigenvalues for concrete operators.

Let ${\mathcal C}[{\mathcal X}]$ be any class of operators on a Banach space ${\mathcal X}$, and let ${Holo}^{-1}({\mathcal C})$ denote the class of operators A for which there exists a holomorphic function f on a neighbourhood ${\mathcal N}$ of the spectrum σ(A) of A such that f is non-constant on connected components of ${\mathcal N}$ and f(A) lies in ${\mathcal C}$. Let ${{\mathcal R}[{\mathcal X}]}$ denote the class of Riesz operators in ${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators $A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$ (the class of all upper or lower [semi] Fredholm operators) by commuting operators in $B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi = 1(B − μi) is Riesz, then there exist decompositions ${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$ and $B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$ such that: (i) If λ ≠ 0, then $\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) if and only if $A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0) $A\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) if and only if $A-B_0\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$), where B0 = ⊕mi = 1(Bi − μi); (iii) if $\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then $A-\lambda B\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$).

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$(X) (resp. ${\mathcal B}$(Y)) be the algebra of all bounded linear operators on X (resp. on Y). For an operator T ∈ ${\mathcal B}$(X) and a vector x ∈ X, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0 ∈X and y0 ∈ Y, we show that a map ϕ from ${\mathcal B}$(X) onto ${\mathcal B}$(Y) satisfies

$ \sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)), $

${\mathcal B}$

In this paper we prove a structure theorem for the class of $\mathcal{AN}$-operators between separable, complex Hilbert spaces which is similar to that of the singular value decomposition of a compact operator. Apart from this, we show that a bounded operator is $\mathcal{AN}$ if and only if it is either compact or a sum of a compact operator and scalar multiple of an isometry satisfying some condition. We obtain characterizations of these operators as a consequence of this structure theorem and deduce several properties which are similar to those of compact operators.

A-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

The ‘polaroid’ property transfers from Banach algebra elements to their tensor product, and hence also to their induced multiplications on ‘ultraprime’ Banach bimodules.

We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.

In this note we answer two question posed by Berkani and Koliha [Acta Sci. Math.69 (2003), 359–376]. We show that generalized Browder's (resp. generalized $a$-Browder's) theorem holds for a Banach space operator if and only if Browder's (resp. $a$-Browder's) theorem does. We also give condition under which generalized Weyl's (resp. generalized $a$-Weyl's) theorem is equivalent to Weyl's (resp. $a$-Weyl's) theorem.

Iseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.