UNBOUNDED OPERATORS ON HILBERT C-MODULES OVER

INTRODUCTION
A (left) Hilbert C-module over a C-algebra A is a left A-moduleW equipped
with an A-valued inner product h
,
i : W W ! A which is A-linear in the
first and conjugate A-linear in the second variable such that W is a Banach space
with the norm kwk = khwjwik1/2.
Throughout this paper we are basically interested in Hilbert modules over
the C-algebras A of compact operators on some complex Hilbert space. These
modules are characterized by the property that each closed submodule is orthogonally
complemented or orthogonally closed (see [5] and [8]). They are generally
not self-dual, i.e. a generalization of Riesz representation theorem for bounded
modular (A-linear) functionals is not valid. Nevertheless, all bounded modular
operators are adjointable.
In the theory of unbounded modular operators densely defined on an arbitrary
Hilbert C-module it is necessary to suppose the regularity condition (see
[4]) on closed operators in order to generalize some basic properties of closed operators
on Hilbert spaces. We point out that the regularity condition is fulfilled
180 BORIS GULJAŠ
for all closed modular operators densely defined on Hilbert C-modules over C-
algebras of compact operators.
It is proved in [2] that the mapping Y : B(W) ! B(We), Y(A) = AjWe , is a
-isomorphism of the C-algebra of all bounded modular operators defined on a
Hilbert module W over the C-algebra of all compact operators on some Hilbert
space and the C-algebra of bounded operators defined on a Hilbert space We
contained in W. The main result in the present paper is the extension of this
mapping to the set of all modular operators densely defined onW. This extension
is an operations preserving bijection from the set of closed modular operators
densely defined in W onto the set of closed operators densely defined on We.
Also, it is a surjective mapping from the set of closable modular operators densely
defined on W onto the set of closable operators densely defined on We. This
mapping enables a natural procedure for lifting the results on densely defined
closed and closable modular operators from Hilbert space theory to Hilbert C-
modules over C-algebras of compact operators.
As application of our technique we prove the polar decomposition of a
closed modular operator as well as the existence of the unique generalized inverse
of a closed modular operator. We also generalize some results on relative
compact operators and closed operators with resolvent in the algebra of compact
operators.
We also note that the technique used in the paper is applicable in Hilbert
H-modules. However, a related discussion on closed and closable operators on
Hilbert H-modules is omitted since the corresponding results are presented in
a similar, or even simpler way. For the results on bounded operators on Hilbert
H-modules we refer to [1] and references therein.