Let H be a Hilbert C-module over a matrix algebra A. It is proved that any function T : H → H which preserves the absolute value of the (generalized) inner product is of the form Tf = φ(f)Uf (f ∈ H), where φ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner’s classical unitary-antiunitary theorem for Hilbert modules. 1991 Physics and Astronomy Cla...

Let H(t) ≥ 1 be a time-dependent self-adjoint operator on a Hilbert space H with quadratic form domain Q(H(t)). If Q(H(t)) is independent of t, along with other suitable conditions, we construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation. We apply this extension of Yosida’s Theorem to study the time evolution of a density matrix in a quantum me...

Exceptional points and double poles of the S matrix are both characterized by the coalescence of a pair of eigenvalues. In the first case, the coalescence causes a defect of the Hilbert space. In the second case, this is not so as shown in previous papers. Mathematically, the reason for this difference is the biorthogonality of the eigenfunctions of a non-Hermitian operator that is ignored in t...

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hardedge of the spectrum of a random Hermitean matrix model. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding ...

The Bogomolny equations for Yang-Mills-Higgs monopoles follow from a system of linear equations which may be solved through a parametric Riemann-Hilbert problem. We extend this approach to noncommutative R and use it to (re)construct noncommutative Dirac, Wu-Yang, and BPS monopole configurations in a unified manner. In all cases we write down the underlying matrix-valued functions for multi-mon...

We consider the Schrödinger operator −∆ − V (x) on R, but with the difference from the usual case that V is a Hermitian matrix-valued potential. In other words, the Hilbert space is not L(R) but L(R;C). The values of functions in this space, ψ(x), are N−dimensional vectors. (What we say here easily generalizes to ‘operatorvalued’ potentials, i.e., C is replaced by a Hilbert space such as L(R), ...

In this paper we extend a result of Šemrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms ...

Abstract. This paper presents a general coding method where data in a Hilbert space are represented by finite dimensional coding vectors. The method is based on empirical risk minimization within a certain class of linear operators, which map the set of coding vectors to the Hilbert space. Two results bounding the expected reconstruction error of the method are derived, which highlight the role...

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, selfadjoint extensions of s...

Let A be an n × n complex matrix such that every row and every column has at most one nonzero entry. We determine permutations of the nonzero entries of A so that the resulting matrix has maximum numerical radius. Extension of the results to operators acting on separable Hilbert spaces are also obtained. Related results and additional problems are also mentioned. AMS Subject Classification. 15A60.