On the spectral multiplicity of a direct sum of operators

Abstract

a nonnegative integer or ∞. Clearly, A is cyclic if and only if μ(A) = 1. The spectral multiplicity is an important invariant of operators, and it plays a key role in operator theory and its applications. Clearly, the notion of cyclic subspace is important in connection with the general problem of existence of a nontrivial invariant subspace, because an operator A ∈ L(X) has no nontrivial invariant subspace if and only if x ∈ Cyc(A) for every x ∈ X\{0}. Cyclic vectors are important in weighted polynomial approximation theory. (More details can be found in [6].) In this article we calculate the spectral multiplicity of the direct sum T ⊕A, where T is a weighted shift operator on a Banach space Y continuously embedded in lp, and A a suitable bounded operator on a Banach space X (Section 2). Note that the main result of Section 2, Theorem 1, generalizes and strengthens some results of the author in [3, Theorem], [4, Theorem 3] and [5, Theorem 3]. First we introduce some notations and definitions. If {ei}i≥0 is a sequence of vectors in a Banach space X, we say that {ei}i≥0 is uniformly minimal if there exists a constant δ > 0 such that