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Norm estimates of discrete Schrödinger operators

Ryszard Szwarc — 1998

Colloquium Mathematicae

Harper’s operator is defined on 2 ( Z ) by H θ ξ ( n ) = ξ ( n + 1 ) + ξ ( n - 1 ) + 2 cos n θ ξ ( n ) , where θ [ 0 , π ] . We show that the norm of H θ is less than or equal to 2 2 for π / 2 θ π . This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.

The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc — 1998

Studia Mathematica

Let G be a group generated by r elements g 1 , , g r . Among the reduced words in g 1 , , g r of length n some, say γ n , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ 2 n has a limit, called the cogrowth exponent with respect to the generators g 1 , , g r . We show by analytic methods that the numbers γ n vary regularly, i.e. the ratio γ 2 n + 2 / γ 2 n is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated...

Structure of geodesics in the Cayley graph of infinite Coxeter groups

Ryszard Szwarc — 2003

Colloquium Mathematicae

Let (W,S) be a Coxeter system such that no two generators in S commute. Assume that the Cayley graph of (W,S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k ≤ d = dist(x,y) there are at most two vertices z such that dist(x,z) = k and dist(z,y) = d - k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from x and y...

An analytic series of irreducible representations of the free group

Ryszard Szwarc — 1988

Annales de l'institut Fourier

Let F k be a free group on k generators. We construct the series of uniformly bounded representations z of F k acting on the common Hilbert space, depending analytically on the complex parameter z, 1 / ( 2 k - 1 ) < | z | < 1 , such that each representation z is irreducible. If z is real or | z | = 1 / ( 2 k - 1 ) then z is unitary; in other cases z cannot be made unitary. For z z ' representations z and z ' are congruent modulo compact operators.

Kaczmarz algorithm in Hilbert space

Rainis HallerRyszard Szwarc — 2005

Studia Mathematica

The aim of the Kaczmarz algorithm is to reconstruct an element in a Hilbert space from data given by inner products of this element with a given sequence of vectors. The main result characterizes sequences of vectors leading to reconstruction of any element in the space. This generalizes some results of Kwapień and Mycielski.

Jacobi matrices on trees

Agnieszka M. KazunRyszard Szwarc — 2010

Colloquium Mathematicae

Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by restriction to the so called radial functions....

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