This monograph has grown out of a recent sequel of jointly written papers by the authors. It proposes to present a unified treatment of the local spectral theory for closed operators acting on a complex Banach space.

While working with closed operators, in a few instances it will be unavoidable to transgress the former concept. More to the point, the theory makes frequent use of operators coinduced by closed operators on a quotient space. Such operators, in general, are not even closable. Special efforts will be requested to exhibit conditions which make them, at least, closable and hence useful instruments in the subsequent theory.

The plan of this work may be sketched as follows. After a brief presentation of the spectral decomposition problem, Chapter I introduces the notion of the single valued extension property with some of its implications. Subsequently, a general study of invariant subspaces is followed by the developing ideas for some special types of subspaces, such as v-spaces, μ-spaces, analytically invariant subspaces, T-absorbent spaces, spectral maximal and T-bounded spectral maximal spaces, T being the given closed operator.

With Chapter II we come to the essence of the spectral theory: the general type of spectral decomposition property with its relationship to the unbounded decomposable operators.

Chapter III is devoted to the spectral duality theory. After having overcome some difficulties due to the general type of the spectral decomposition problem, it was gratifying to have obtained a spectral duality theorem, under the natural constraints of the problem.