Pick Interpolation and Hilbert Function Spaces

Abstract

§3.3. H p spaces again §3.4. #°°(ID>) as a multiplier algebra §3.5. Inner functions 45 §3.6. Historical notes Chapter 4. P 2 (/_i) 49 §4.1. Other spaces with H°°(TD>) as the multiplier algebra 49 §4.2. Vector-valued-P 2 (/u) spaces Chapter 5. Pick Redux §5.1. Necessity of positivity of the Pick matrix §5.2. The Szego kernel has the Pick property §5.3. The Caratheodory problem §5.4. Uniqueness of the Szego kernel §5.5. Historical notes Chapter 6. Qualitative Properties of the Solution of the Pick Problem in H°°(B) §6.1. A formula for the solution §6.2. The realization formula for H°°(B) §6.3. Another formula for the solution §6.4. The Nevanlinna problem Chapter 7. Characterizing Kernels with the Complete Pick Property §7.1. Characterization of the complete Pick property §7.2. Another characterization of the complete Pick property §7.3. Holomorphic spaces with the complete Pick property §7.4. The Sobolev space §7.5. The M sxt Pick property §7.6. Historical notes Contents xi Chapter 8. The Universal Pick Kernel §8.1. The universal kernel 97 §8.2. The realization formula for the universal kernel §8.3. Qualitative properties of solutions of the Pick problem for complete Pick kernels §8.4. The Toeplitz-corona theorem §8.5. Beurling theorems §8.6. Holomorphic complete Pick spaces §8.7. The Nevanlinna problem §8.8. Uniqueness of kernels with the Pick property §8.9. Historical notes Chapter 9.

Hilbert spaces of Dirichlet series and their multipliers

“…See [2] for a treatment of complete Pick kernels. For every integer n ≥ 2, let F (n) be the number of ways n can be factored into a product of integers greater than 1, where the order matters (so e.g.…”

“…Theorem 3.4 is a special case of the realization formula for general complete Pick kernels (see [4], [3], [18], [2]), and we omit the proof.…”

“…[13] or [2]). Therefore, every sequence {λ i } ∞ i=1 that tends to the line {σ = ρ} has a subsequence that is an interpolating sequence for H w : just choose the n th element in the subsequence so that the first n − 1 entries in the n th row and n th column of G are smaller in modulus than 2 −n , and then the off-diagonal entries will have Hilbert-Schmidt norm smaller than 1.…”

“…It is a theorem of D. Marshall and C. Sundberg [12] that, for any space with the Pick property, the interpolating sequences for the Hilbert space and for its multiplier algebra coincide (the proof can also be found in [2]). Coupling this with the previous observation, we conclude that for every boundary point of Ω ρ , there is a sequence that converges to that boundary point and that is interpolating for M ult(H w ).…”

“…[2] for a discussion). We show that, unlike for the spaces H α , the multipliers do not extend to be analytic on a larger domain than the domain of analyticity of the Hilbert space, by constructing an interpolating sequence for the multiplier algebra that accumulates on the boundary of the halfplan of convergence of the Hilbert space (Theorem 3.6).…”