View source for Operator vessel

The theory of operator vessels provides a framework for the [[Spectral analysis|spectral analysis]] and synthesis of tuples of commuting non-self-adjoint (or non-unitary) operators, especially for operators that are not  "too far"  from being self-adjoint (or unitary). It reveals deep connections with [[Algebraic geometry|algebraic geometry]], especially with function theory on a compact real Riemann surface (i.e., a compact Riemann surface endowed with an anti-holomorphic involution, cf. also [[Riemann surface|Riemann surface]]), and with the theory of multi-dimensional systems. The theory of operator vessels can be also generalized for the study of tuples of non-commuting non-self-adjoint (or non-unitary) operators (cf. also [[Non-self-adjoint operator|Non-self-adjoint operator]]), especially for non-self-adjoint representations of Lie algebras (or non-unitary representations of discrete groups).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300602.png" /> be a pair of commuting bounded linear operators in a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300603.png" />. A quasi-Hermitian commutative two-operator vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300604.png" /> is a collection

<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/o130060/o1300605.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300606.png" /> is an auxiliary Hilbert space called the external space of the vessel (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300607.png" /> is called the internal space), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300608.png" /> is a bounded linear mapping, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o1300609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006012.png" /> are bounded self-adjoint operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006013.png" /> (cf. also [[Self-adjoint operator|Self-adjoint operator]]), such that

<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/o130060/o13006014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<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/o130060/o13006015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

<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/o130060/o13006016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<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/o130060/o13006017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

The equations (a2) are so-called colligation conditions (or node conditions) well-known from the spectral analysis of a single non-self-adjoint operator (cf. also [[Operator colligation|Operator colligation]]); they allow one to  "isolate"  the non-Hermitian parts of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006019.png" />. The equations (a3), (a4) and (a5) are deeper; the self-adjoint operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006021.png" /> carry information about the interaction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006023.png" />.

Notice that a given pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006025.png" /> of commuting operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006026.png" /> can be always embedded in a quasi-Hermitian commutative vessel by setting

<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/o130060/o13006027.png" /></td> </tr></table>

<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/o130060/o13006028.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006029.png" /> denotes the orthogonal projection on the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006030.png" />; it is an easy consequence of the commutativity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006032.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006033.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006035.png" />).

The notion of an operator vessel allows one to construct commuting operators with more complicated spectral data out of commuting operators with simpler spectral data while controlling the non-Hermitian parts. Let

<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/o130060/o13006036.png" /></td> </tr></table>

<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/o130060/o13006037.png" /></td> </tr></table>

be two quasi-Hermitian commutative vessels with the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006038.png" />. Using the coupling construction familiar in the spectral analysis of a single non-self-adjoint operator, one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006039.png" /> and defines operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006041.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/o130060/o13006042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

<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/o130060/o13006043.png" /></td> </tr></table>

(the operators being written in block form with respect to the direct sum decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006044.png" />). It is clear that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006046.png" /> satisfy the colligation conditions:

<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/o130060/o13006047.png" /></td> </tr></table>

but in general they do not commute. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006048.png" /> exactly when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006049.png" />; more precisely,

<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/o130060/o13006050.png" /></td> </tr></table>

(with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006052.png" />) is a quasi-Hermitian commutative vessel if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006053.png" /> (the matching theorem).

Assume now that the external space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006054.png" /> of a vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006055.png" /> as in (a1) is finite dimensional; one defines a polynomial in two complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006057.png" /> by setting

<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/o130060/o13006058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006059.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006061.png" /> is a polynomial with real coefficients of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006062.png" />. One calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006063.png" /> the discriminant polynomial of the vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006064.png" />, and the projective [[Plane real algebraic curve|plane real algebraic curve]] with an affine equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006065.png" /> — the discriminant curve. A generalized Cayley–Hamilton theorem holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006066.png" /> (under the natural minimality assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006067.png" />). It follows that the joint spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006069.png" /> (cf. also [[Taylor joint spectrum|Taylor joint spectrum]]) lies on the (affine part of the) discriminant curve.

The following remarkable equality also holds:

<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/o130060/o13006070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

Thus, associated to the vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006071.png" /> there is the discriminant polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006072.png" />, and two self-adjoint determinantal representations of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006074.png" />, called (for system-theoretic reasons, see below) the input and the output determinantal representations, respectively.

Consider now the inverse problem of constructing, up to unitary equivalence, all (minimal) quasi-Hermitian commutative two-operator vessels with given discriminant polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006075.png" />, given input determinantal representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006076.png" />, and with the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006078.png" /> in the vessel having given joint spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006079.png" />, which is a subset of the affine part of the real projective plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006080.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006081.png" />. Here, two vessels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006082.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006083.png" />) are said to be unitarily equivalent if there is an [[Isometric mapping|isometric mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006084.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006085.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006086.png" /> such that

<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/o130060/o13006087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>

<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/o130060/o13006088.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006089.png" /> is an [[Irreducible polynomial|irreducible polynomial]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006090.png" /> is a smooth irreducible curve (these assumptions can be relaxed), a complete and explicit solution of the inverse problem stated above has been obtained. This solution leads to triangular models for the corresponding pair of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006092.png" /> with finite non-Hermitian ranks, similar to the well-known triangular models for a single non-self-adjoint operator. The solution is based on first constructing elementary objects — vessels with one-dimensional internal space corresponding to the points of the joint spectrum — and then coupling them using the matching theorem. It follows from (a5) that in a vessel with one-dimensional internal space the output determinantal representation is determined by the input determinantal representation and the spectral data; the successive matching of output and input determinantal representations in the matching theorem then gives a system of non-linear difference (for the discrete part of the spectrum) and differential (for the continuous part of the spectrum) equations for self-adjoint determinantal representations of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006093.png" />. The algebro-geometric assumptions on the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006094.png" /> and the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006095.png" /> imply that self-adjoint determinantal representations are naturally parametrized by certain points in the [[Jacobi variety|Jacobi variety]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006096.png" />; and it turns out that passing from a self-adjoint determinantal representation to the corresponding point in the Jacobi variety linearizes the systems of non-linear difference and differential equations alluded to above. Actually, the system can even be solved explicitly using theta-functions, yielding explicit formulas for the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006098.png" /> in a triangular model.

The fundamental interplay between the spectral theory of a pair of commuting non-self-adjoint operators with finite non-Hermitian ranks and function theory on a compact real [[Riemann surface|Riemann surface]] is based on the notion of the joint characteristic function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o13006099.png" /> be a quasi-Hermitian commutative vessel as in (a1), with discriminant polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060100.png" /> and discriminant curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060101.png" />. For each affine point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060103.png" />, one may define two non-trivial subspaces of the external space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060104.png" />:

<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/o130060/o130060105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>

<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/o130060/o130060106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>

Then for arbitrary complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060108.png" /> (such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060109.png" /> is outside the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060110.png" />), the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060111.png" />,

<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/o130060/o130060112.png" /></td> </tr></table>

maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060113.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060114.png" /> and the restriction of this operator to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060115.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060117.png" />. The joint characteristic function of the vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060118.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/o130060/o130060119.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>

<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/o130060/o130060120.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060121.png" /> is an affine point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060122.png" /> outside the joint spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060124.png" />.

The joint characteristic function is thus a mapping of certain sheaves on the discriminant curve (cf. also [[Sheaf|Sheaf]]). For simplicity, assume that the discriminant polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060125.png" /> has only one, possibly multiple, irreducible factor; thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060126.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060127.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060128.png" /> is the irreducible affine equation of the discriminant curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060129.png" />. Assume also that both the input and the output determinantal representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060130.png" /> are maximal, meaning that for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060131.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060132.png" /> the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060134.png" /> have maximal possible dimension (which is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060135.png" /> times the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060136.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060137.png" />; notice that all these assumptions are trivially satisfied when the discriminant polynomial is irreducible, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060138.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060139.png" /> is a smooth irreducible curve). It follows then that the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060141.png" /> for different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060142.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060143.png" /> (including, of course, the points at infinity) fit together to form two complex holomorphic rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060144.png" /> vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060146.png" /> on a compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060147.png" /> which is the desingularization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060148.png" /> (cf. also [[Resolution of singularities|Resolution of singularities]]). The joint characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060149.png" /> (naturally extended to be identity at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060150.png" /> at infinity) is simply a bundle mapping, holomorphic outside the joint spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060152.png" />. Notice that since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060153.png" /> is a real curve, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060154.png" /> is a real Riemann surface, that is, a Riemann surface equipped with an anti-holomorphic involution (the complex conjugation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060155.png" />).

Assuming the maximality of the input and the output determinantal representations, the joint characteristic function of a (minimal) vessel determines the vessel uniquely up to unitary equivalence. The joint characteristic function is expansive with respect to certain naturally defined scalar products on the vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060156.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060157.png" />. Conversely, given any bundle mapping between the kernel vector bundles corresponding to the given two maximal self-adjoint determinantal representations, which is expansive with respect to the corresponding scalar products, this bundle mapping can be realized as the joint characteristic function of a quasi-Hermitian commutative vessel with these input and output determinantal representations.

Kernel vector bundles corresponding to maximal self-adjoint determinantal representations are isomorphic (up to an inessential twist) to vector bundles of multiplicative half-order differentials, i.e., to vector bundles of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060158.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060159.png" />, the canonical line bundle (the line bundle of holomorphic differentials), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060160.png" /> is a flat vector bundle associated to some representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060161.png" /> of the fundamental group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060162.png" />. Using this isomorphism one may replace the joint characteristic function by the so-called normalized joint characteristic function, which is simply a mapping of flat vector bundles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060163.png" />, i.e., a multiplicative multi-valued matrix function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060164.png" /> (with appropriate matrix multipliers on the left and on the right). The normalized joint characteristic function is usually more convenient for analytic investigations. There are also functional models for the corresponding pair of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060166.png" /> with finite non-Hermitian ranks, similar to the well-known functional models of Sz.-Nagy–Foias and de Branges–Rovnyak for a single operator; the model space is an appropriately defined space of multiplicative half-order differentials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060167.png" />, and the model operators are certain  "compressed multiplication operators"  by the affine coordinate functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060169.png" />.

Like the notion of colligation in the spectral theory of a single non-self-adjoint operator (cf. also [[Operator colligation|Operator colligation]]), the notion of a vessel has a system-theoretic significance. Given a quasi-Hermitian commutative two-operator vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060170.png" /> as in (a1), one writes a linear shift-invariant continuous two-dimensional system

<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/o130060/o130060171.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>

<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/o130060/o130060172.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>

<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/o130060/o130060173.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060174.png" /> is the state with values in the internal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060176.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060177.png" /> are, respectively, the input and the output with values in the external space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060178.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060179.png" />. The colligation conditions (a2) imply that the system (a13)–(a15) satisfies the energy balance law:

<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/o130060/o130060180.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>

<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/o130060/o130060181.png" /></td> </tr></table>

for any direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060182.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060183.png" />. Unlike the usual one-dimensional systems, the system (a13)–(a15) is overdetermined (cf. also [[Overdetermined system|Overdetermined system]]), the compatibility conditions arising from the equality of mixed partial derivatives:

<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/o130060/o130060184.png" /></td> </tr></table>

The commutativity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060185.png" /> means precisely that the system is consistent for an arbitrary initial state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060186.png" /> and the identically zero input. The vessel condition (a3) implies that a sufficient (and under some assumptions also a necessary) condition for the input signal to be compatible is given 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/o130060/o130060187.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>

The vessel conditions (a4), (a5) imply that the corresponding output satisfies

<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/o130060/o130060188.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a18)</td></tr></table>

The joint characteristic function of the vessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060189.png" /> is the so-called joint [[Transfer function|transfer function]] of the overdetermined system (a13)–(a15) together with the compatibility partial differential equations (a17) and (a18) at the input and at the output, respectively.

The notion of a quasi-Hermitian commutative two-operator vessel is the simplest and the best studied; it can be successfully generalized in various directions, like:

1) Quasi-Hermitian commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060190.png" />-operator vessels for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060191.png" />, which give a framework for the spectral analysis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060192.png" />-tuples of commuting non-self-adjoint operators (especially with finite non-Hermitian ranks).

2) Quasi-unitary commutative operator vessels, which give a framework for the spectral analysis of tuples of commuting non-unitary operators (especially with finite defects); they are related to discrete conservative multi-dimensional systems (rather than continuous).

3)  "Non-metric"  commutative operator vessels, which correspond to overdetermined multi-dimensional systems together with compatibility partial differential equations at the input and at the output, but without any energy balance laws.

4) Non-commutative generalizations, in particular (quasi-Hermitian)  "Lie algebra"  vessels, where one replaces a tuple of commuting operators by a representation of a given [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060193.png" />. Such vessels provide a framework for the spectral analysis of non-self-adjoint representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060194.png" />. The associated (conservative) multi-dimensional system evolves on a [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060195.png" /> having the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130060/o130060196.png" />.

The theory of operator vessels was initiated by M.S. Livšic [[#References|[a2]]], [[#References|[a3]]]. The term  "vessel"  was coined in the book [[#References|[a1]]]; earlier papers use the term  "regular colligation" . The book [[#References|[a1]]] provides a comprehensive treatment of the subject. A shorter survey, containing also the more recent results, is [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</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">[a2]</TD> <TD valign="top">  M.S. Livšic,   "Operator waves in Hilbert space and related partial differential equations"  ''Integral Eq. Oper. Th.'' , '''2''' :  1  (1979)  pp. 25–47</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.S. Livšic,   "A method for constructing triangular canonical models of commuting operators based on connections with algebraic curves"  ''Integral Eq. Oper. Th.'' , '''3''' :  4  (1980)  pp. 489–507</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V. Vinnikov,   "Commuting operators and function theory on a Riemann surface"  S. Axler (ed.)  J. McCarthy (ed.)  D. Sarason (ed.) , ''Holomorphic Spaces and Their Operators'' , ''Math. Sci. Res. Inst. Publ.'' , '''33''' , Cambridge Univ. Press  (1998)  pp. 445–476</TD></TR></table>