All unitary (contractive) perturbations of a given unitary operator
by
finite-rank-
operators with fixed range can be parametrized by
unitary (contractive)
matrices
; this generalizes
unitary rank-one ()
perturbations, where the Aleksandrov–Clark family of unitary
perturbations is parametrized by the scalars on the unit circle
.
For a strict contraction
the
resulting perturbed operator
is (under the natural assumption about star cyclicity of the range) a completely
nonunitary contraction, so it admits the functional model.
We investigate the Clark operator, i.e., a unitary operator that intertwines
(written in the spectral representation of the nonperturbed operator
) and
its model. We make no assumptions on the spectral type of the unitary operator
; an
absolutely continuous spectrum may be present.
We first find a universal representation of the adjoint Clark operator in the
coordinate-free Nikolski–Vasyunin functional model; the word “universal” means
that it is valid in any transcription of the model. This representation can
be considered to be a special version of the vector-valued Cauchy integral
operator.
Combining the theory of singular integral operators with the theory of functional
models, we derive from this abstract representation a concrete formula for the adjoint
of the Clark operator in the Sz.-Nagy–Foiaş transcription. As in the scalar case, the
adjoint Clark operator is given by a sum of two terms: one is given by the boundary
values of the vector-valued Cauchy transform (postmultiplied by a matrix-valued
function) and the second one is just the multiplication operator by a matrix-valued
function.
Finally, we present formulas for the direct Clark operator in the Sz.-Nagy–Foiaş
transcription.
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