Contraction (operator theory)
contracting operator, contractive operator, compression
A bounded linear mapping of a Hilbert space
into a Hilbert space
with
. For
, a contractive operator
is called completely non-unitary if it is not a unitary operator on any
-reducing subspace different from
. Such are, for example, the one-sided shifts (in contrast to the two-sided shifts, which are unitary). Associated with each contractive operator
on
there is a unique orthogonal decomposition,
, into
-reducing subspaces such that
is unitary and
is completely non-unitary.
is called the canonical decomposition of
.
A dilation of a given contractive operator acting on is a bounded operator
acting on some large Hilbert space
such that
,
where
is the orthogonal projection of
onto
. Every contractive operator in a Hilbert space
has a unitary dilation
on a space
, which, moreover, is minimal in the sense that
is the closed linear span of
(the Szökefalvi-Nagy theorem). Minimal unitary dilations and functions of them, defined via spectral theory, allow one to construct a functional calculus for contractive operators. This has been done essentially for bounded analytic functions in the open unit disc
(the Hardy class
). A completely non-unitary contractive operator
belongs, by definition, to the class
if there is a function
,
, such that
. The class
is contained in the class
of contractive operators
for which
,
as
. For every contractive operator of class
there is the so-called minimal function
(that is, an inner function
,
in
,
almost-everywhere on the boundary of
) such that
and
is a divisor of all other inner functions with the same property. The set of zeros of the minimal function
of a contractive operator
in
, together with the complement in the unit circle of the union of the arcs along which
can be analytically continued, coincides with the spectrum
. The notion of a minimal function of a contractive operator
of class
allows one to extend the functional calculus for this class of contractive operators to certain meromorphic functions in
.
The theorem on unitary dilations has been obtained not only for individual contractive operators but also for discrete, ,
and continuous,
,
, semi-groups of contractive operators.
As for dissipative operators (cf. Dissipative operator), also for contractive operators a theory of characteristic operator-valued functions has been constructed and, on the basis of this, also a functional model, which allows one to study the structure of contractive operators and the relations between the spectrum, the minimal function and the characteristic function (see [1]). By the Cayley transformation
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a contractive operator is related to a maximal accretive operator
, that is,
is such that
is a maximal dissipative operator. Constructed on this basis is the theory of dissipative extensions
of symmetric operators
(respectively, Philips dissipative extensions
of conservative operators
).
The theories of similarity, quasi-similarity and unicellularity have been developed for contractive operators. The theory of contractive operators is closely connected with the prediction theory of stationary stochastic processes and scattering theory. In particular, the Lax–Philips scheme [2] can be considered as a continual analogue of the Szökefalvi-Nagy–Foias theory of contractive operators of class .
References
[1] | B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators in Hilbert space" , North-Holland (1970) (Translated from French) |
[2] | P.D. Lax, R.S. Philips, "Scattering theory" , Acad. Press (1967) |
Comments
A reducing subspace for an operator is a closed subspace
such that there is a complement
, i.e.
, such that both
and
are invariant under
, i.e.
,
.
References
[a1] | I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian) |
[a2] | I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc. (1970) (Translated from Russian) |
Contraction (operator theory). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contraction_(operator_theory)&oldid=19347