A Local Spectral Theory for Closed Operators

Abstract

This book, which is almost entirely devoted to unbounded operators, gives a unified treatment of the contemporary local spectral theory for unbounded closed operators on a complex Banach space. While the main part of the book is original, necessary background materials provided. There are some completely new topics treated, such as the complete spectral duality theory with the first comprehensive proof of the predual theorem, in two different versions. Also covered are spectral resolvents of various kinds (monotomic, strongly monotonic, almost localized, analytically invariant), and spectral decompositions with respect to the identity. The book concludes with an extensive reference list, including many papers published in the People's Republic of China, here brought to the attention of Western mathematicians for the first time. Pure mathematicians, especially those working in operator theory and functional analysis, will find this book of interest.

Stability of the local spectrum

“…The function x T is called the local resolvent function of T at x. See [2], [3] and [6] for further details.…”

“…In [2], Erdelyi and Lange prove that if T is an operator satisfying the Single Valued Extension Property (hereafter referred to as SVEP) and x T is the local resolvent function of T in x, then σ( x T (λ), T ) = σ(x, T ) (2) for all λ ∈ C \ σ(x, T ). Moreover, if A is an operator which commutes with an operator T satisfying the SVEP, then σ(Ax, T ) ⊂ σ(x, T ), (3) for all x ∈ X. In particular, if A has an inverse, then the expression (3) turns into an equality.…”

Equivalent conditions to the spectral decomposition property for closed operators

“…For a subset Y of X, Y denotes the annihilator of Y in X* and for Zd',we use the symbol XZ for the preannihilator of Z in X. For the rest, the terminology and notation conform to that employed in [3].…”

“…This paper gives some necessary and sufficient conditions for a closed operator to possess the spectral decomposition property.In the monograph [3] and in a sequel of papers by the authors, a local spectral theory has been built for closed operators on the sole assumption of the spectral decomposition property. As an abstraction of Dunford's concept of "spectral reduction ' [2, p. 1927] and that of Bishop's "duality theory of type 3" [1], an operator T endowed with the spectral decomposition property produces a spectral decomposition of the underlying space, pertinent to any finite open cover of the spectrum o(T).…”

“…For a subset Y of X, Y denotes the annihilator of Y in X* and for Zd',we use the symbol XZ for the preannihilator of Z in X. For the rest, the terminology and notation conform to that employed in [3].We shall adopt and adjust some concepts and ideas from Bishop's "duality theory of type 4" [1, §4]. A couple Ux and U2 of a bounded and an unbounded Cauchy domain, related by U2 = (Ux )c, are referred to as complementary simple sets.…”