For which reproducing kernel Hilbert spaces is Pick's theorem true?

Pick's theorem tells us that there exists a function inH , which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH ², and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.     

Grauert- and Lax-Halmos-type theorems and extension of matrices with entries in H

In the paper we prove an extension theorem for matrices with entries in H^∞(U) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.     

Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces

Let H be a separable complex Hilbert space, A a von Neumann algebra in ?(H),a faithful, normal state on A. We prove that if a sequence (X_n: n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σ^∞_n=1n⁻² Var(X_n) log²(n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L₂=L₂(A,φ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.     

Nevanlinna-Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna-Pick interpolation problem is studied in the class Sκ of meromorphic functions f with κ poles inside the unit disk D and with ‖f_‖L^∞(T)?1. In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna-Pick interpolation conditions and has the minimal possible L^∞-norm on the unit circle T.     

Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p  -Laplacian (1<p<∞$1<p<\infty$) obtained by Matei (2000) [19] and Takeuchi (1998) [22], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet p-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa (2013) [9].     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

On a result of G. Brosch

We prove a uniqueness theorem for non-constant meromorphic functions f, g which share three values 0, 1, ∞ and f−a, g−b share the value 0 for a,b∉{0,1,∞}. Our theorem improves a result of G. Brosch.     

Kharitonov-like theorems for robust performance of interval systems

The sensitivity function of a control system is an important concept in performance analysis, classical filter design as well as modern robust H^∞ control. For an interval system, we prove that the maximal H^∞ norm of its sensitivity function is achieved at twelve (out of sixteen) Kharitonov vertices. Similar Kharitonov-like results are established for the complementary sensitivity function and strict positive realness of interval systems. These results are useful in robust performance analysis and H_∞ control design for dynamic systems under parametric perturbations.     

Proper asymptotic unitary equivalence in KK-theory and projection lifting from the corona algebra

In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using KK-theory and prove a result which asserts that there is a unitary of the form ‘identity + compact’ which gives the unitary equivalence of two projections if the ‘essential codimension’ of two projections vanishes for certain C^∗-algebras employing the proper asymptotic unitary equivalence of KK-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of C(X)⊗B where X is [0,1], (−∞,∞), [0,∞), and [0,1]/{0,1}.     

Some remarks on the free interpolation by bounded and slowly increasing analytic functions

One constructs a new linear bounded operator which solves the problem of free interpolation in the Hardy space H. This operator is analogous to the well-known interpolation operator of P. W. Jones. One of the fundamental distinctions of the operator constructed in this paper consists in the fact that it acts in the smallest (in a certain sense) linear closed subspace of the space H which, in turn, gives the possibility to obtain new results on free interpolation for several subclasses of the spaces H. In the second part of the paper one proves a theorem which, in a significant number of cases, allows us to reduce the solution of the problem on multiple interpolation (when the multiplicities are bounded in their totality) to the problem of free interpolation with simple nodes. One gives several examples of the application of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 35–46, 1983.     

Duality, Tangential Interpolation, and Töplitz Corona Problems

In this paper, we extend a method of Arveson (J Funct Anal 20(3):208–233, 1975) and McCullough (J Funct Anal 135(1):93–131, 1996) to prove a tangential interpolation theorem for subalgebras of H ^∞. This tangential interpolation result implies a Töplitz corona theorem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent and Wick (Complex Anal Oper Theory 3(3):729–738, 2009) and Douglas and Sarkar (Proc CRM, 2009).     

The topology of flows near a dicritical singularity

In this paper, we prove a topological finite determinacy theorem for a generic family of C^∞ vector fields at a dicritical singularity in any dimension.     

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c^∞(G)-C_c^∞(H)-bimodule C_c^∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c^∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

OnH 2 minimization of theH ∞ suboptimal solutions

In this paper we show that theH ² minimization of theH suboptimal solutions for a class of suboptimalH distance problems can be reduced to a finite dimensional nonlinear optimization problem. This extends a result of [7] where the same problem is considered in the Caratheodory-Schur interpolation case.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.