Ritz values and the free joint numerical radius
Paul Reine Kennett Dela Rosa, Hugo J. Woerdeman · 2021
Let A [is an element of] C^[n [times] n] be normal. For a given z in the numerical range of A, consider the set B_[A,k](z) of k [times] k matrices W for which [matrix [z * / 0 W]] is a compression of A. Elements of B_[A,1](z) are the Ritz values associated to z. If no eigenvalues of A are in the interior of its numerical range, we identify the smallest convex region containing B_[A,1](z). If no three eigenvalues of A lie on the same line, we also prove that for any compact subset K of the numerical range of A that avoids the eigenvalues, the mappings z [maps to] B_[A,k](z) and z [maps to] [sigma](B_[A,k](z)) are uniformly continuous on K. The free joint numerical radius w(X₁, ... , X_m) of Hilbert space operators X₁, ... , X_m [is an element of] B(H) is w(X₁, ... , X_m) = sup{w(X₁ [o times] U₁ + ... + X_m [o times] U_m)}, where the supremum is taken over every Hilbert space K, every choice of m unitaries U₁, ... , U_m [is an element of] B(K), and the tensor product is spatial. The free joint numerical radius coincides with the classical numerical radius when there is only one operator (m = 1). We prove a formula for the free joint numerical radius of a tuple of generalized permutations. When dim(H) < [infinity], we show that w(X₁, ... ,X_m) < 1/2 is equivalent to the existence of a fixed point of the operator-valued function f_[X₁, ... ,X_m](Z) := I + [sum]_[j=1]^m [(Z^[1/2]X_j^*ZX_jZ^[1/2] + [1/4]I)^[1/2] + (Z^[1/2]X_jZX_j^*Z^[1/2] + [1/4]I)^[1/2]]. We also present a conjecture how to compute such fixed point.