Extensions of positive definite functions on amenable groups

Let $S$ be a subset of a amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of the paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as a corollary. New applications are also presented

Two remarks about nilpotent operators of order two

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.Comment: 7 pages. To appear in Proceedings of the AM

Redheffer Products and Characteristic Functions

AbstractFuhrmann [Israel J. Math.16 (1973), 162-176], and subsequently Ball and Lubin [Pacific J. Math.63 (1976), 309-324] have studied a class of perturbations of completely nonunitary contractions. We extend their results concerning the computation of the characteristic function by using the "Redheffer product" machinery [J. Math. Phys.39 (1960), 269-286]. This has been familiar to system theory experts for many years and has been recently revived by Foias and Frazho ["The Commutant Lifting Approach to Interpolation Problems," Birkhäuser, Basel, 1990] to obtain alternate proofs in the theory of intertwining dilations of contractions on a Hilbert space. The proof obtained is conceptually surprisingly simple. An application is the recapture, from a point of view different from the original one, of a result concerning de Branges′ spaces, which have received renewed attention in recent years

The Central Ando Dilation and Related Orthogonality Properties

AbstractA pair of commuting contractions on a Hilbert space has in general many nonisomorphic minimal unitary dilations. We show that there is a “distinguished” one. This can be viewed in many respects as an analogue of the central intertwining lifting in the family of all intertwining liftings of a contraction

Extensions of positive definite functions on free groups

AbstractAn analogue of Krein's extension theorem is proved for operator-valued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization of noncommutative polynomials