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Principal currents for a pair of unitary operators. (English) Zbl 0810.47018

Mem. Am. Math. Soc. 522, 103 p. (1994).
Let \(A\) be a self-adjoint \(n\times n\) complex matrix and let \(v\) be a non-zero vector of \(\mathbb{C}^ n\). How is the spectrum \(\lambda_ j(B)\) of the non-negative rank-one perturbation \(B= A+ v\otimes v\) related to the spectrum \(\lambda_ j(A)\) of \(A\)? A simple analysis (using for instance Courant’s minimax principle) shows that: \[ \lambda_ 1(B)\geq \lambda_ 1(A)\geq \lambda_ 2(B)\geq\cdots\geq \lambda_ n(B)\geq \lambda_ n(A). \] The union of the spectral displacement intervals \(J= [\lambda_ 1(A),\lambda_ 1(B)]\cup\cdots\cup [\lambda_ n(A),\lambda_ n(B)]\) satisfies the following remarkable identity (which can be left as an exercise to the reader): \[ \text{det}\{(B- z)(A- z)^{-1}\}= \exp\left(\int_ J {dt\over t-z}\right),\quad (\text{Im } z>0).\tag{1} \] The left hand side function in (1) is called the perturbation determinant of the pair \((A,B)\). The exponential of the Cauchy transform in the right hand side is a typical rational function \(R(z)\) belonging to the Nevanlinna class (\(R(\infty)=1\) and \(\text{Im }R(z)>0\) for \(\text{Im }z> 0\)).
It was M. G. Krein [Math. Sb. 33, 597-626 (1953; Zbl 0052.123); Dokl. Akad. Nauk SSSR 144, 268-271 (1962; Zbl 0191.152)] who has forseen the potential of formula (1) and has generalized it to trace class perturbations of Hilbert space self-adjoint operators. More precisely, for \(A\) and \(B\) self-adjoint and with \(\text{Tr}| A- B|< \infty\), one finds: \[ \text{det}\{(B- z)(A- z)^{-1}= \exp\left(\int_{\mathbb{R}} {\delta(t)dt\over t- z}\right),\quad (\text{Im } z>0).\tag{2} \] The new parameter in this formula is the phase shift \(\delta\in L^ 1(\mathbb{R})\). For about three decades, M. G. Krein and his school have developed an effective, faithful dictionary between the spectral properties of the pair \((A,B)\) and the function theory properties of \(\delta\). For more details, see M. G. Krein, “Topics in differential and integral equations and operator theory”, Boston (1983; Zbl 0512.45001).
The two-dimensional analogues of Krein’s phase-shift and the perturbation determinant formula (2) were discovered in the late sixties and early seventies by the first author. His subsequent joint work with R. W. Carey has expanded in a variety of directions and unexpected applications. See R. W. Carey and the first author [J. Funct. Anal. 19, 50-80 (1975; Zbl 0309.47026)] and the references contained in the present Memoir. For a bounded, linear Hilbert space operator \(T\) with trace-class self- commutator \((\text{Tr}|[T^*,T]|< \infty)\) the analogue of Krein’s formula reads: \[ \text{det}\{(T- z)^{*-1}(T- w)(T-z)^*(T- w)^{-1}\}=\exp\left(-\pi^{-1} \int_{\mathbb{C}} {g(u)d_ A(u)\over (u- z)(\bar u-\bar w)}\right),\tag{3} \] where \(z\) and \(w\) are large complex numbers and \(dA\) stands for the planar Lebesgue measure. The corresponding spectral parameter \(g\in L^ 1(\mathbb{C})\) is called the principal function of the operator \(T\), or equivalently, of the pair of self-adjoint operators \((\text{Re }T,\text{Im }T)\).
A vast program of revealing the correspondence between spectral properties of the operator \(T\) and measure theory or even more subtle metric properties of the principal function \(g\) was carried out in the last two decades by Pincus and his collaborators and a few other authors. Due to the high technicality and interdisciplinarity of these researches, the theory of the principal function was, and it is still, very slowly assimilated by the mathematical community. Needless to say that the cyclic cohomology has its roots in some trace formulae arising in the theory of the principal function.
The Memoir under review represents a culmination of the efforts of Pincus and his school in achieving their program. The main object of study here is a pair of unitary operators with rank-one difference. In this case the principal function is defined naturally on the two-dimensional torus. An ingenious Hilbert space geometry construction combined with methods of scattering theory and singular integral equations produces two more functions besides the principal function of the original unitary pair. The thesis of the Memoir, well illustrated by deep results, is that in these three functions one can read the refined spectral structure of the original operators. Unfortunately the details are too ample to be even sketched in this review.
The new non-commutative spectral analysis proposed by Pincus and Zhou in this Memoir can be traced back to the classical work of H. Weyl on the Sturm-Liouville equation and on the other hand it is naturally connected to other branches of mathematical analysis, such as the dilation theory of Hilbert space contractions, scattering theory, geometric integration theory and maybe in the future, as the authors mention in the introduction, to algebraic \(K\)-theory. Adjacent topics (too briefly treated in the text, but not less important) are the similarity theory of Toeplitz operators and the spectral theory of linear differential operators.
The Memoir certainly represents a mature, profoundly original and important contribution to modern operator theory. Its presentation bears the personal, Wagnerian style of Pincus. (This is a warning for the new reader to approach the book as a new opus of the maestro from Bayreuth).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A55 Perturbation theory of linear operators
46L45 Decomposition theory for \(C^*\)-algebras
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47A40 Scattering theory of linear operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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