A bounded
linear operator
,
acting on a
Hilbert space,
with the property that its
self-commutator
is
trace-class,
i.e.
.
A semi-normal operator can equivalently be defined by a pair
of self-adjoint operators (cf.
Self-adjoint operator)
with trace-class commutator (after writing
).
The theory
of semi-normal operators is one of the few well-developed
spectral theories
for a class of non-self-adjoint operators. For the
latter, see
[a3].
Most examples of semi-normal operators are obtained via the
Berger–Shaw inequality:
If
is a
hyponormal operator
(i.e.
),
of finite rational cyclicity
,
then:
where
is the spectrum of
(cf. also
Spectrum of an operator).
Here, the
rational cyclicity
(also called the
rational multiplicity)
of an operator
is defined as follows. Let
be the spectrum of
and let
be the algebra of rational functions of a complex variable with poles
outside
.
Then
is the smallest cardinal number such that there is a set of vectors
such that the closure of the span of
is the whole space.
If
,
then
is said to be
rationally cyclic.
In particular, normal and
rationally cyclic subnormal operators are semi-normal,
[a2].
Certain singular
integral operators with Cauchy-type kernel are also
semi-normal, see
[a3],
[a4],
[a6]
and
Singular integral.
One of the most refined unitary invariants of a pure
semi-normal operator
is the
principal function
,
which was
introduced by
J.D. Pincus
[a6].
Let
,
be polynomials in two complex variables, and let
be the
Jacobian
of the pair
,
written in complex coordinates. The
Pincus–Helton–Howe trace formula
[a4]
characterizes
:
Via this formula one can prove the functoriality of the principal
function
under the
holomorphic functional calculus.
An observation due to
D. Voiculescu
shows that
is invariant under
Hilbert–Schmidt perturbations
of
.
The entire behaviour of the principal function qualifies it as the
correct two-dimensional analogue of
Krein's spectral shift function,
well known in the
perturbation theory
of
self-adjoint operators,
see
[a5].
The above trace formula can be interpreted as a
generalized index theorem;
this was one of the origins
cyclic cohomology.
Thanks to a deep result of
T. Kato
and
C.R. Putnam,
a pure hyponormal
operator
with trace-class self-commutator has
absolutely continuous real and imaginary parts
and
.
Consequently, by diagonalizing
on a vector-valued
-space
supported on the real line, the operator
becomes:
where
,
are essentially bounded operator-valued functions.
This
singular integral model
extends to all semi-normal operators;
it was the source of most
results in this area, by putting together methods of
scattering theory
(cf. also
Scattering matrix)
and
singular integral equations,
cf.
[a1],
[a7].
Several invariant-subspace results are known for semi-normal
operators. For instance,
S.W. Brown
has shown that hyponormal operators with
thick spectrum
(that is, dominant spectrum in an open subset of
)
have non-trivial invariant subspaces. As an application
of the theory of the principal function,
C.A. Berger
has proved that sufficiently
high powers of hyponormal operators have invariant subspaces. For both
results see
[a5].
One of the most studied sets of semi-normal operators is the
class of operators
with rank-one self-commutator:
.
For irreducible
,
the
determinantal function
[a6]:
is a complete unitary invariant of
.
This function can be expressed as the exponential of a double
Cauchy transform
of the principal function
.
A variety of applications of the above determinantal function to
inverse problems of potential theory in the plane are known,
[a1],
[a5].