A spectral commutant lifting theorem

Abstract

Abstract.The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.

Spectral versus classical Nevanlinna-Pick interpolation in dimension two

“…|F (λ)| sp ≤ 1 for λ ∈ D. A result analogous to Pick's theorem was proved in [3], and it involves the positivity of a matrix constructed from data W j similar to W j , i.e. W j = X j W j X −1 j for invertible operators X j ∈ M N (C).…”

The two-by-two spectral Nevanlinna-Pick problem

“…, n, and r(F (λ)) ≤ 1 for all λ ∈ D. Here r( ) denotes the spectral radius of a matrix. The problem has attracted much attention over the past 20 years [6,7,8,16,13,15] partly because it is a challenging variant of a well-loved classical topic, but mainly because it is a test case of a fundamental question which arises in H ∞ control, the problem of µ-synthesis [10,11]. A solution of the general problem would have applications to the design of automatic controllers that are robust with respect to structured uncertainty.…”

“…The spectral Nevanlinna-Pick problem is alluringly simple in its formulation, but far-reaching investigations by Bercovici, Foiaş, and Tannenbaum [6,7,8] reveal that it contains a great deal of subtlety. For example, one of the thrusts of their work is to show by some ingenious examples [6] that the idea of diagonalisation, or "interpolating the eigenvalues", does not resolve the problem.…”

“…For example, one of the thrusts of their work is to show by some ingenious examples [6] that the idea of diagonalisation, or "interpolating the eigenvalues", does not resolve the problem. They obtain qualitative results on extremals, and they use a variant of commutant lifting theory to prove a theorem [6,Theorem 4] …”

On the zero set of the Kobayashi–Royden pseudometric of the spectral unit ball

“…, A N ∈ Ω n decide whether there is a mapping ϕ ∈ O(D, Ω n ) such that ϕ(a j ) = A j , 1 ≤ j ≤ N (cf. [1,2,4,7,8] The infinitesimal version of the above is the Carathéodory-Fejér problem of order 1: given matrices A 0 , A 1 ∈ M n , decide whether there is a mapping ϕ ∈ O(D, Ω n ) such that A 0 = ϕ(0), A 1 = ϕ ′ (0). This problem has been studied in [10].…”