An operator satisfying Dunford's condition (C) but without bishop's property (β)

For X a complex Banach space and U an open subset of the complex plane С, let O (U, X) denote the space of analytic X- valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping TU on O(U, X) given by (Tuf)(λ) = (λ – T)f(λ) for every f e O(U, X) and λ e U. Corresponding to each closed F ⊂ С there is also an associated analytic subspace XT(F) = X ∩ ran(7c//F). For an arbitrary T e L(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.

Strongly analytic spaces in spectral decomposition

It is now well-known that decomposable operators have a rich structure theory; in particular, an operator is decomposable iff its adjoint is [3]. There are many other criteria for decomposability [8], [9]. In Theorem 2.2 of this paper (see below) we give several new ones. Some of these (e.g. (ii), (iii)) are “relaxations” of conditions given in [7] and [8]. Assertion (vi) is a version of a result in [10]. Characterizations (iv)and (v) are novel in two respects. For instance, (v) states that an operator Tcan be “patched” together into a decomposable operator if it has an invariant subspace Y such that T | Y and the coinduced operator T | Y are both decomposable. Secondly, in this way the strongly analytic subspace appears in the theory of spectral decomposition.