Wierd Hilbert space problem

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

  1. Piotr Soltan

    Piotr Soltan Guest

    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

    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
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