Interpolating Matrices
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Integral Equations and Operator Theory
Interpolating Matrices Alberto Dayan Abstract. We extend Carleson’s interpolation theorem to sequences of matrices, by giving necessary and sufficient separation conditions for a sequence of matrices to be interpolating. Keywords. Interpolating sequences, Corona theorem, Model spaces, Feichtinger conjecture.
1. Introduction A sequence Λ = (λn )n∈N of points in the unit disc D is interpolating for the space H∞ of bounded analytic functions on D if for any bounded sequence (wn )n∈N there exists a non zero function f in H∞ such that f (λn ) = wn , for any n in N. Intuitively, being interpolating is a matter of how separated the sequence is in D. Since the interpolating functions are holomorphic a natural way to compute distances between points of Λ is by using the pseudohyperbolic distance z1 − z2 . ρ(z1 , z2 ) := |bz1 (z2 )| = 1 − z1 z2 Here bτ will denote the involutive Blaschke factor at a point τ in D. We will say that Λ is strongly separated if ρ(λn , λk ) > 0. inf n∈N
(1.1)
k=n
A celebrated result due to Carleson [2] affirms that Λ is interpolating if and only if it is strongly separated. For a geometric proof, see [7, Th. 1.1, Ch. VII]. In [1, Th. 9.42], one can find the same result restated (and proved) in terms of separation conditions on the reproducing kernel functions (kλn )n∈N in H2 . We use both those approaches to characterize interpolating sequences of matrices (of any dimensions) with spectra in the unit disk. Throughout the discussion, a holomorphic functions f will be applied to a square matrix M via the Riesz–Dunford functional calculus. Section 3 will make it precise, but at this stage just observe that, since we will consider 0123456789().: V,-vol
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functions holomorphic in the unit disc, the eigenvalues of all the matrices we will consider are in D. A first attempt to emulate the definition of an interpolating sequence of scalars can then be to define a sequence (An )n∈N of square matrices with spectra in D to be interpolating if, given a sequence (Wn )n∈N of square matrices which is bounded in the operator norm, then there exists a function f in H∞ such that f (An ) = Wn . One reason why this can’t be the right approach is given by the following example: let Λ = (λn )n∈N be a Blaschke sequence in D, that is, (1 − |λn |) < ∞, n∈N
and set
λ An = n 0
1 . λn
The least that we can expect from a definition of interpolating sequence of matrices that is consistent with the scalar case is that (f (An ))n∈N is a target sequence too, for any f in H∞ . But if we set B to be the Blaschke product at the points of Λ, then (B (λn ))n∈N is unbounded, and so is 0 B (λn ) B(An ) = . 0 0 Therefore, a target sequence for an interpolating sequence of matrices can’t simply be a sequence of matrices bounded with respect the operator norm. The problem seems to be that, given a Jordan block λ 1 J= λ∈D 0 λ and a bounded holomorphic function f , then f (λ) f (λ) f (J) = , 0 f (λ) and while f (λ) can assume any
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