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''contracting operator, contractive operator, compression''
 
''contracting operator, contractive operator, compression''
  
A bounded linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258301.png" /> of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258302.png" /> into a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258303.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258304.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258305.png" />, a contractive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258306.png" /> is called completely non-unitary if it is not a unitary operator on any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258307.png" />-reducing subspace different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258308.png" />. Such are, for example, the one-sided shifts (in contrast to the two-sided shifts, which are unitary). Associated with each contractive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c0258309.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583010.png" /> there is a unique orthogonal decomposition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583011.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583012.png" />-reducing subspaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583013.png" /> is unitary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583014.png" /> is completely non-unitary. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583015.png" /> is called the canonical decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583016.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583017.png" /> is a bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583018.png" /> acting on some large Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583021.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583022.png" /> is the orthogonal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583023.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583024.png" />. Every contractive operator in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583025.png" /> has a unitary dilation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583026.png" /> on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583027.png" />, which, moreover, is minimal in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583028.png" /> is the closed linear span of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583029.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583030.png" /> (the Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583031.png" />). A completely non-unitary contractive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583032.png" /> belongs, by definition, to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583033.png" /> if there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583035.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583036.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583037.png" /> is contained in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583038.png" /> of contractive operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583039.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583041.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583042.png" />. For every contractive operator of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583043.png" /> there is the so-called minimal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583044.png" /> (that is, an inner function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583048.png" /> almost-everywhere on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583049.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583051.png" /> is a divisor of all other inner functions with the same property. The set of zeros of the minimal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583052.png" /> of a contractive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583053.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583054.png" />, together with the complement in the unit circle of the union of the arcs along which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583055.png" /> can be analytically continued, coincides with the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583056.png" />. The notion of a minimal function of a contractive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583057.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583058.png" /> allows one to extend the functional calculus for this class of contractive operators to certain meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583059.png" />.
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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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583061.png" /> and continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583063.png" />, semi-groups of contractive operators.
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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|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 [[#References|[1]]]). By the Cayley transformation
 
As for dissipative operators (cf. [[Dissipative operator|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 [[#References|[1]]]). By the Cayley transformation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583064.png" /></td> </tr></table>
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\begin{equation*} A = ( I + T ) ( I - T ) ^ { - 1 } , \quad 1 \notin \sigma _ { p } ( T ), \end{equation*}
  
a contractive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583065.png" /> is related to a maximal accretive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583066.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583067.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583068.png" /> is a maximal dissipative operator. Constructed on this basis is the theory of dissipative extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583069.png" /> of symmetric operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583070.png" /> (respectively, Philips dissipative extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583071.png" /> of conservative operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583072.png" />).
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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 [[#References|[2]]] can be considered as a continual analogue of the Szökefalvi-Nagy–Foias theory of contractive operators of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583073.png" />.
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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 [[#References|[2]]] can be considered as a continual analogue of the Szökefalvi-Nagy–Foias theory of contractive operators of class $C_{00}$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  Ch. Foiaş,  "Harmonic analysis of operators in Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.D. Lax,  R.S. Philips,  "Scattering theory" , Acad. Press  (1967)</TD></TR></table>
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<table><tr><td valign="top">[1]</td> <td valign="top">  B. Szökefalvi-Nagy,  Ch. Foiaş,  "Harmonic analysis of operators in Hilbert space" , North-Holland  (1970)  (Translated from French)</td></tr><tr><td valign="top">[2]</td> <td valign="top">  P.D. Lax,  R.S. Philips,  "Scattering theory" , Acad. Press  (1967)</td></tr></table>
  
  
  
 
====Comments====
 
====Comments====
A reducing subspace for an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583074.png" /> is a closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583075.png" /> such that there is a complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583076.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583077.png" />, such that both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583079.png" /> are invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583080.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025830/c02583082.png" />.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  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)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  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)</td></tr></table>

Revision as of 16:55, 1 July 2020

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)
How to Cite This Entry:
Contraction (operator theory). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contraction_(operator_theory)&oldid=50093
This article was adapted from an original article by I.S. Iokhvidov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article