A classical theorem of Pick gives a criterion for interpolation by analytic functions in the open unit disc D subject to an H∞-norm bound. This result has substantial generalizations in two different directions. On the one hand, one can replace H∞ by the multiplier algebras of certain Hilbert function spaces, some of them having no connection with analyticity [Ag1, Ag2]. On the other hand, one ...

Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the " thermodynamic limit " of the spectral shift function per u...

A similar concentration result is available for subgaussian random variables. It is known as the Hanson-Wright inequality and is given in Proposition 2 below. First versions of this inequality can be found in Hanson and Wright [5] and Wright [9], although with a weaker statement than Proposition 2 below since these results involve ||| (|aij |) |||2 instead of |||A|||2. Recent proofs of this con...

The Hahn-Banach theorem in its simplest form asserts that a bounded linear functional defined on a subspace of a Banach space can be extended to a linear functional defined everywhere, without increasing its norm. There is an order-theoretic version of this extension theorem (Theorem 0.1 below) that is often more useful in context. The purpose of these lecture notes is to discuss the noncommuta...

It is proved that a discrete group G is amenable if and only if for every unitary representation of G in an infinite-dimensional Hilbert space H the maximal uniform compactification of the unit sphere SH has a G-fixed point, that is, the pair (SH, G) has the concentration property in the sense of Milman. Consequently, the maximal U(H)equivariant compactification of the sphere in a Hilbert space...

New unique characterization results for the potential V (x) in connection with Schrödinger operators on R and on the half-line [0,∞) are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary sp...

Abstract. In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of squa...

We first extend properties of the Fuglede-Kadison determinant on a finite von Neumann algebra M to L p (M), any p > 0. Using this we give several useful variants of the noncommutative Szegö theorem for L p (M), including the one usually attributed to Kolmogorov and Krein. As applications, we generalize the noncommutative Jensen inequality, and generalize many of the classical results concerning...

The Friedrichs extension and the Krein extension of a positive operator in a Krein space are characterized in terms of their spectral functions in a Krein space.