View source for Semi-normal operator

A bounded [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200901.png" />, acting on a [[Hilbert space|Hilbert space]], with the property that its self-commutator is trace-class, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200902.png" />. A semi-normal operator can equivalently be defined by a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200903.png" /> of self-adjoint operators (cf. [[Self-adjoint operator|Self-adjoint operator]]) with trace-class commutator (after writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200904.png" />). The theory of semi-normal operators is one of the few well-developed spectral theories for a class of non-self-adjoint operators. For the latter, see [[#References|[a3]]].

Most examples of semi-normal operators are obtained via the Berger–Shaw inequality: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200905.png" /> is a hyponormal operator (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200906.png" />), of finite rational cyclicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200907.png" />, then:

<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/s/s120/s120090/s1200908.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s1200909.png" /> is the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009010.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]).

Here, the rational cyclicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009011.png" /> (also called the rational multiplicity) of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009012.png" /> is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009013.png" /> be the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009014.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009015.png" /> be the algebra of rational functions of a complex variable with poles outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009016.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009017.png" /> is the smallest cardinal number such that there is a set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009018.png" /> such that the closure of the span of

<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/s/s120/s120090/s12009019.png" /></td> </tr></table>

is the whole space.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009021.png" /> is said to be rationally cyclic. In particular, normal and rationally cyclic subnormal operators are semi-normal, [[#References|[a2]]]. Certain singular integral operators with Cauchy-type kernel are also semi-normal, see [[#References|[a3]]], [[#References|[a4]]], [[#References|[a6]]] and [[Singular integral|Singular integral]].

One of the most refined unitary invariants of a pure semi-normal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009022.png" /> is the principal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009023.png" />, which was introduced by J.D. Pincus [[#References|[a6]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009025.png" /> be polynomials in two complex variables, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009026.png" /> be the Jacobian of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009027.png" />, written in complex coordinates. The Pincus–Helton–Howe trace formula [[#References|[a4]]] characterizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009028.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/s/s120/s120090/s12009029.png" /></td> </tr></table>

Via this formula one can prove the functoriality of the principal function under the holomorphic functional calculus. An observation due to D. Voiculescu shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009030.png" /> is invariant under Hilbert–Schmidt perturbations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009031.png" />. The entire behaviour of the principal function qualifies it as the correct two-dimensional analogue of Krein's spectral shift function, well known in the [[Perturbation theory|perturbation theory]] of self-adjoint operators, see [[#References|[a5]]]. The above trace formula can be interpreted as a generalized index theorem; this was one of the origins [[Cyclic cohomology|cyclic cohomology]].

Thanks to a deep result of T. Kato and C.R. Putnam, a pure hyponormal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009032.png" /> with trace-class self-commutator has absolutely continuous real and imaginary parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009034.png" />. Consequently, by diagonalizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009035.png" /> on a vector-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009036.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009037.png" /> supported on the real line, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009038.png" /> becomes:

<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/s/s120/s120090/s12009039.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/s/s120/s120090/s12009040.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009042.png" /> are essentially bounded operator-valued functions. This singular integral model extends to all semi-normal operators; it was the source of most results in this area, by putting together methods of scattering theory (cf. also [[Scattering matrix|Scattering matrix]]) and singular integral equations, cf. [[#References|[a1]]], [[#References|[a7]]].

Several invariant-subspace results are known for semi-normal operators. For instance, S.W. Brown has shown that hyponormal operators with thick spectrum (that is, dominant spectrum in an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009043.png" />) have non-trivial invariant subspaces. As an application of the theory of the principal function, C.A. Berger has proved that sufficiently high powers of hyponormal operators have invariant subspaces. For both results see [[#References|[a5]]].

One of the most studied sets of semi-normal operators is the class of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009044.png" /> with rank-one self-commutator: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009045.png" />. For irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009046.png" />, the determinantal function [[#References|[a6]]]:

<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/s/s120/s120090/s12009047.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/s/s120/s120090/s12009048.png" /></td> </tr></table>

is a complete unitary invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009049.png" />. This function can be expressed as the exponential of a double Cauchy transform of the principal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120090/s12009050.png" />. A variety of applications of the above determinantal function to inverse problems of potential theory in the plane are known, [[#References|[a1]]], [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Clancey,   "Seminormal operators" , ''Lecture Notes Math.'' , '''742''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Conway,   "Theory of subnormal operators" , ''Math. Surveys Monogr.'' , '''36''' , Amer. Math. Soc.  (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Gohberg,   S. Goldberg,   M.A. Kaashoeck,   "Classes of linear operators I–II" , Birkhäuser  (1990/3)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.W. Helton,   R. Howe,   "Traces of commutators of integral operators"  ''Acta Math.'' , '''135'''  (1975)  pp. 271–305</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Martin,   M. Putinar,   "Lectures on hyponormal operators" , Birkhäuser  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.D. Pincus,   "Commutators and systems of singular integral equations I"  ''Acta Math.'' , '''121'''  (1968)  pp. 219–249</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Xia,   "Spectral theory of hyponormal operators" , Birkhäuser  (1983)</TD></TR></table>