View source for Operator colligation

''colligation of operators, node of operators, operator node''

An aggregate of spaces and operators, to which is associated a characteristic operator-valued function. This characteristic function reflects properties of the colligation; for instance, multiplication of colligations results in multiplication of the corresponding characteristic functions. Colligations are used to build models for individual operators, but they are also employed in extension theory, interpolation theory, factorization problems, scattering theory, and system theory. They appear in different forms; the most frequently encountered forms are discussed below.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300501.png" /> be a bounded linear mapping in a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300502.png" />, and assume that the imaginary part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300504.png" /> is represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300506.png" /> is a bounded linear mapping from a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300507.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o1300509.png" /> is a bounded, self-adjoint and unitary operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005010.png" /> (cf. also [[Linear operator|Linear operator]]; [[Self-adjoint operator|Self-adjoint operator]]; [[Unitary operator|Unitary operator]]). The aggregate of spaces and operators

<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/o/o130/o130050/o13005011.png" /></td> </tr></table>

is called an operator colligation, and the corresponding operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005012.png" /> acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005013.png" /> and defined by

<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/o/o130/o130050/o13005014.png" /></td> </tr></table>

is called the characteristic operator-valued function of the colligation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005015.png" />. Clearly, the characteristic function satisfies the identity

<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/o/o130/o130050/o13005016.png" /></td> </tr></table>

which exhibits many properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005017.png" />. The present notion of colligation is due to M.S. Brodskii and M.S. Livšic [[#References|[a6]]], while the characteristic function was introduced, in a slightly different form, by Livšic in [[#References|[a9]]]. The characteristic function is a unitary invariant of the [[Non-self-adjoint operator|non-self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005018.png" />, and the singular points of this function coincide with the spectrum of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005019.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005020.png" /> is a powerful tool for the investigation of the spectral properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005021.png" /> (invariant subspaces, triangular and functional models, Jordan representations, characterization of spectra, similarity), but it also plays a role in complex analysis (factorization into Blaschke–Potapov factors, interpolation problems); see [[#References|[a3]]], [[#References|[a17]]]. In system theory, the characteristic function is interpreted as the operator-valued [[Transfer function|transfer function]] of the system (conservative in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005022.png" />) of the form

<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/o/o130/o130050/o13005023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005024.png" /> is an input vector, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005025.png" /> is an output vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005026.png" /> is a state space vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005027.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005028.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005029.png" /> is determined through the imaginary part of the non-self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005030.png" />, and therefore this function is responsible for the spectral analysis of operators  "close"  to bounded self-adjoint operators.

In the case where the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005031.png" /> is unbounded, the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005032.png" /> is assumed to be rigged, i.e. there exists a triplet of Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005033.png" /> (cf. also [[Rigged Hilbert space|Rigged Hilbert space]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005034.png" /> be a bounded linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005035.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005038.png" />. In this case, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005040.png" /> is a bounded linear operator from a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005041.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005043.png" /> is a bounded, self-adjoint, and unitary operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005044.png" />. Now the aggregate of spaces and operators

<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/o/o130/o130050/o13005045.png" /></td> </tr></table>

is called a rigged operator colligation and the corresponding characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005046.png" />, acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005047.png" />, is defined by

<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/o/o130/o130050/o13005048.png" /></td> </tr></table>

In this triplet setting, the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005050.png" /> (conservative in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005051.png" />) with an input vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005052.png" />, an output vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005053.png" /> and a state space vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005054.png" />, leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005055.png" />. Clearly, the definition for the unbounded case preserves the algebraic formalism developed for the bounded case. Rigged operator colligations and their characteristic operator-valued functions have been introduced by E.R. Tsekanovskii and Yu.L. Shmulyan [[#References|[a18]]]. These types of colligations (with bounded or unbounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005056.png" />) appear for instance in the theory of circuits, systems with distributed parameters and in scattering theory, see [[#References|[a10]]], [[#References|[a12]]]. Different definitions of characteristic functions (without colligations) for unbounded non-self-adjoint operators have been introduced and studied in [[#References|[a8]]], [[#References|[a15]]], [[#References|[a16]]].

The above colligations are associated to operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005057.png" /> which are  "close"  to being self-adjoint. Similar notions can be developed for operators which are  "close"  to being unitary.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005058.png" /> be a bounded operator in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005059.png" />. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005061.png" /> be Hilbert spaces, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005064.png" /> be bounded linear operators. The aggregate of spaces and operators

<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/o/o130/o130050/o13005065.png" /></td> </tr></table>

is called a unitary colligation if the operator matrix

<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/o/o130/o130050/o13005066.png" /></td> </tr></table>

is a unitary operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005067.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005068.png" />. The corresponding characteristic operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005069.png" />, acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005070.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005071.png" />, is defined on the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005072.png" /> by

<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/o/o130/o130050/o13005073.png" /></td> </tr></table>

whose values are contractive inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005074.png" /> and unitary on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005075.png" />, as can be seen from identities such as

<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/o/o130/o130050/o13005076.png" /></td> </tr></table>

The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005077.png" /> is called the main operator of the colligation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005078.png" /> and it is a [[Contraction(2)|contraction]]. Conversely, any contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005079.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005080.png" /> is the main operator of some unitary colligation, since the operator matrix

<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/o/o130/o130050/o13005081.png" /></td> </tr></table>

is unitary. The spectral theory of operators  "close"  to unitary operators has been developed by B. Sz.-Nagy and C. Foiaş [[#References|[a13]]] through the characteristic functions of these augmented contractions. Later the theory was reformulated in terms of unitary operator colligations and their characteristic functions, see [[#References|[a4]]]. The colligation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005082.png" /> is called isometric, or co-isometric, if the operator matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005083.png" /> is isometric, respectively co-isometric. Unitary, isometric, or co-isometric colligations and their characteristic functions are being used in model theory [[#References|[a19]]], [[#References|[a14]]], and interpolation problems in Schur classes (cf. [[Schur functions in complex function theory|Schur functions in complex function theory]]); they also appear in extension theory, scattering theory and system theory [[#References|[a2]]]. For instance, a discrete-time system of the form

<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/o/o130/o130050/o13005084.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005088.png" />, is called conservative if the corresponding operator colligation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005089.png" /> is unitary. Solving this system by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005092.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005096.png" />, leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005097.png" />.

In all types of colligations questions arise concerning the dimension of the state space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130050/o13005098.png" /> and the minimality of the representation. Likewise, there are various constructions to build state spaces by means of characteristic functions (model theory). Operator colligations have been considered in the setting of Banach spaces [[#References|[a5]]] and in the setting of indefinite inner product spaces (see for instance [[#References|[a1]]] for Pontryagin spaces (cf. also [[Pontryagin space|Pontryagin space]]), and [[#References|[a1]]], p. 205, for references for the case of Krein spaces, cf. also [[Krein space|Krein space]]). Recently (1999) there is interest in colligations associated with commuting operators [[#References|[a11]]] and in colligations with several variables [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Alpay,   A. Dijksma,   J. Rovnyak,   H.S.V. de Snoo,   "Schur functions, operator colligations, and reproducing kernel Pontryagin spaces" , ''Oper. Th. Adv. Appl.'' , '''96''' , Birkhäuser  (1997)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.Z. Arov,   L.Z. Grossman,   "Scattering matrices in the theory of extensions of isometric operators"  ''Math. Nachr.'' , '''157'''  (1992)  pp. 105–123</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.S. Brodskii,   "Triangular and Jordan representations of linear operators" , ''Transl. Math. Monographs'' , '''32''' , Amer. Math. Soc.  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.S. Brodskii,   "Unitary operator colligations and their characteristic functions"  ''Russian Math. Surveys'' , '''33''' :  4  (1978)  pp. 159–191  ''Uspekhi Mat. Nauk.'' , '''33''' :  4 (202)  (1978)  pp. 141–168</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Bart,   I. Gohberg,   M.A. Kaashoek,   "Minimal factorization of matrix and operator functions" , ''Oper. Th. Adv. Appl.'' , '''1''' , Birkhäuser  (1979)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.S. Brodskii,   M.S. Livšic,   "Spectral analysis of non-selfadjoint operators and intermediate systems"  ''Amer. Math. Soc. Transl.'' , '''13''' :  2  (1960)  pp. 265–346  ''Uspekhi Mat. Nauk.'' , '''13''' :  1 (79)  (1958)  pp. 3–85</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.A. Ball,   T.T. Trent,   "Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna–Pick interpolation in several variables"  ''J. Funct. Anal.'' , '''157'''  (1998)  pp. 1–61</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Kuzhel,   "Characteristic functions and models of nonselfadjoint operators" , Kluwer Acad. Publ.  (1996)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.S. Livšic,   "On the spectral decomposition of linear non-selfadjoint operators"  ''Amer. Math. Soc. Transl.'' , '''5''' :  2  (1957)  pp. 67–114  ''Mat. Sb.'' , '''34''' :  76  (1954)  pp. 145–199</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M.S. Livšic,   "Operators, oscillations, waves" , ''Transl. Math. Monographs'' , '''34''' , Amer. Math. Soc.  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M.S. Livšic,   N. Kravitsky,   A.S. Markus,   V. Vinnikov,   "Theory of commuting nonselfadjoint operators" , Kluwer Acad. Publ.  (1995)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M.S. Livšic,   A.A. Yantsevich,   "Operator colligations in Hilbert spaces" , Winston  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  B. Sz.-Nagy,   C. Foiaş,   "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  N. Nikolski,   V. Vasyunin,   "Elements of spectral theory in terms of the free function model I. Basic constructions" , ''Holomorphic spaces (Berkeley, CA, 1995)'' , Cambridge Univ. Press  (1998)  pp. 211–302</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  B.S. Pavlov,   "Spectral analysis of a singular Schrödinger operator in terms of a functional model" , ''Partial Differential Equations VIII'' , Springer  (1995)  pp. 89–153</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A.V. Štrauss,   "Characteristic functions of linear operators"  ''Amer. Math. Soc. Transl.'' , '''40''' :  2  (1964)  pp. 1–37  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''24'''  (1960)  pp. 43–74</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  L.A. Sakhnovich,   "Interpolation theory and its applications" , Kluwer Acad. Publ.  (1997)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  E.R. Tsekanovskii,   Yu.L. Shmulyan,   "The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator colligations and characteristic functions"  ''Russian Math. Surveys'' , '''32''' :  5  (1977)  pp. 73–131  ''Uspekhi Mat. Nauk.'' , '''32''' :  5  (1977)  pp. 69–124</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  L. de Branges,   J. Rovnyak,   "Square summable power series" , Holt, Rinehart &amp; Winston  (1966)</TD></TR></table>