# Wierd Hilbert space problem

Discussion in 'Math Research' started by Piotr Soltan, Nov 9, 2003.

1. ### Piotr SoltanGuest

I would appreciate any comment/reference for the following problem:

On a Hilbert space we are given a closed densely defined operator x,
positive selfadjoint non degenerate operator Q. We know that there are
many (I will explain what I mean by "many" below) bounded operators y
such that

a) Q^{-1}yQ extends to a bounded operator,
b) xy extends to a bounded operator,
c) Q^{-1}xyQ extends to a bounded operator.

I would like to know if I can expect (perhaps there are some
conditions for this) that there exist many bounded operators z such
that

a') QzQ^{-1} extends to a bounded operator,
b') x^*z extends to a bounded operator,
c') Qx^*zQ^{-1} extends to a bounded operator.

In the original problem the operators y and z were to be chosen from
some fairly arbitrary non degenerate C*-subalgebra of B(H), so by
"many" I mean for example that among them there is a net strongly
convergent to the identity operator and they are not compact.

I'd appreciate any remarks

Piotr Soltan

Piotr Soltan, Nov 9, 2003