Analytic Tridiagonal Reproducing Kernels

Abstract

Abstract. This paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(a n,0 + a n,1 z + · · · + a n,J z J )z n , n ≥ 0}.The primary focus is on the tridiagonal case where J = 1 and how it compares to the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated and aspects of this operator are discussed.In the diagonal case Mz is a weighted unilateral shift. It is shown that in the tridiagonal case this need not be so and an example is given in which the commutant of Mz on a tridiagonal space is strikingly different from that on any diagonal space.

Linear dynamics in reproducing kernel Hilbert spaces

“…Then M * z on H(k) is hypercyclic if lim inf n 1 (n!) 2 In the Hilbert space setting, this result unifies previous work on sufficient conditions for hypercyclicity of weighted shift operators (see Gethner and Shapiro [17], Kitai [21], Rolewicz [26], and Salas [27]).…”

“…Choose two sequences of non-zero complex numbers µ = {µ n } and ν = {ν n } such that {e n } n≥0 is an orthonormal basis of a reproducing kernel Hilbert space (of analytic functions on D) H µ,ν (see Theorem 1 in [2]), where e n (z) = µ n z n + ν n z n+1 (z ∈ D, n ≥ 0).…”

“…Since k µ,ν (z, w) = n≥0 e n (z)e n (w), we get the tridiagonal kernel as (cf. [2]) where a = |µ 0 | 2 and b = µ 0 ν 0 . Fix n ≥ 0 and set…”

Multishifts on directed Cartesian products of rooted directed trees

“…Hence (v (1) , v (2) ) ∈ E implies that v (2) ∈ Chi(v (1) ). Similarly, (v (2) , v (3) ) ∈ E implies that v (3) ∈ Chi(v (2) ) ⊆ Chi 2 (v (1) ). Consequently, v (n) ∈ Chi n−1 (v (1) ).…”

“…To see (iii), suppose there is a finite sequence {w (i) } n i=1 (n 2) of distinct vertices such that (w (i) , w (i+1) ) ∈ E ⊗ for all 1 i n − 1 and (w (n) , w (1) ) ∈ E ⊗ . Then w (1) j ∈ Chi n (w (1) j ) for all 1 j d. This is a contradiction to the fact that T j has no circuits.…”