SPECTRAL REPRESENTATION OF ABSOLUTELY MINIMUM ATTAINING UNBOUNDED NORMAL OPERATORS

Let T : D(T) -> H-2 be a densely defined closed operator with domain D(T) subset of H-1. We say T to be absolutely minimum attaining if for every non-zero closed subspace M of H-1 with D(T)boolean AND M not equal {0}, the restriction operator T|(M) : D(T)boolean AND M -> H-2 attains its minimum modulus m(T|(M)). That is, there exists x is an element of D(T)boolean AND M with ||x|| = 1 and ||T(x)|| = inf{||T(m)|| : m is an element of D(T) boolean AND M : ||m|| = 1}. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. One such important characterization is that an unbounded operator belongs to this class if and only if its null space is finite dimensional and its Moore-Penrose inverse is compact. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent.