Contraction (operator theory)

contracting operator, contractive operator, compression

A bounded linear mapping $T$ of a Hilbert space $H$ into a Hilbert space $H _ { 1 }$ with $\| T \| \leq 1$. For $H = H _ { 1 }$, a contractive operator $T$ is called completely non-unitary if it is not a unitary operator on any $T$-reducing subspace different from $\{ 0 \}$. Such are, for example, the one-sided shifts (in contrast to the two-sided shifts, which are unitary). Associated with each contractive operator $T$ on $H$ there is a unique orthogonal decomposition, $H = H _ { 0 } \otimes H _ { 1 }$, into $T$-reducing subspaces such that $T _ { 0 } = T | _ { H _ { 0 } }$ is unitary and $T _ { 1 } = T | _ { H _ { 1 } }$ is completely non-unitary. $T = T _ { 0 } \otimes T _ { 1 }$ is called the canonical decomposition of $T$.

A dilation of a given contractive operator acting on $H$ is a bounded operator $B$ acting on some large Hilbert space $K \supset H$ such that $T ^ { n } = P B ^ { n }$, $n = 1,2 , \dots,$ where $P$ is the orthogonal projection of $K$ onto $H$. Every contractive operator in a Hilbert space $H$ has a unitary dilation $U$ on a space $K \supset H$, which, moreover, is minimal in the sense that $K$ is the closed linear span of $\{ U ^ { n } H \} _ { n = - \infty } ^ { + \infty }$ (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 $D$ (the Hardy class $H ^ { \infty }$). A completely non-unitary contractive operator $T$ belongs, by definition, to the class $C _ { 0 }$ if there is a function $u \in H ^ { \infty }$, $u ( \lambda ) \not \equiv 0$, such that $u ( T ) = 0$. The class $C _ { 0 }$ is contained in the class $C_{00}$ of contractive operators $T$ for which $T ^ { n } \rightarrow 0$, $T ^ { * n } \rightarrow 0$ as $n \rightarrow \infty$. For every contractive operator of class $C _ { 0 }$ there is the so-called minimal function $m _ { T } ( \lambda )$ (that is, an inner function $u \in H ^ { \infty }$, $| u ( \lambda ) | \leq 1$ in $D$, $| u ( e ^ { i t } ) | = 1$ almost-everywhere on the boundary of $D$) such that $m _ { T } ( T ) = 0$ and $m _ { T } ( \lambda )$ is a divisor of all other inner functions with the same property. The set of zeros of the minimal function $m _ { T } ( \lambda )$ of a contractive operator $T$ in $D$, together with the complement in the unit circle of the union of the arcs along which $m _ { T } ( \lambda )$ can be analytically continued, coincides with the spectrum $\sigma ( T )$. The notion of a minimal function of a contractive operator $T$ of class $C _ { 0 }$ allows one to extend the functional calculus for this class of contractive operators to certain meromorphic functions in $D$.

The theorem on unitary dilations has been obtained not only for individual contractive operators but also for discrete, $\{ T ^ { n } \}$, $n = 0,1 , \ldots,$ and continuous, $\{ T ( s ) \}$, $0 \leq s \leq \infty$, 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

\begin{equation*} A = ( I + T ) ( I - T ) ^ { - 1 } , \quad 1 \notin \sigma _ { p } ( T ), \end{equation*}

a contractive operator $T$ is related to a maximal accretive operator $A$, that is, $A$ is such that $i A$ is a maximal dissipative operator. Constructed on this basis is the theory of dissipative extensions $B_0$ of symmetric operators $A _ { 0 }$ (respectively, Philips dissipative extensions $i B _ { 0 }$ of conservative operators $i A _ { 0 }$).

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 $C_{00}$.

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 $T$ is a closed subspace $K$ such that there is a complement $K ^ { \prime }$, i.e. $H = K \oplus K ^ { \prime }$, such that both $K$ and $K ^ { \prime }$ are invariant under $T$, i.e. $T ( K ) \subset K$, $T ( K ^ { \prime } ) \subset K ^ { \prime }$.

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)