James Rovnyak

Schur functions, operator colligations, and reproducing kernel Pontryagin spaces

Using the theory of linear relations in Pontryagin spaces we extend to the nonpositive case the theory of reproducing kernel spaces associated with contractions in Hilbert spaces.

The factorization problem for nonnegative operator valued functions

Introduction. Let f be a function defined on the circle T = {e : 0 ̂ 6 < 2T } or line R = ( — <*>, <*> ) whose values are nonnegative operators on a separable complex Hubert space. We are concerned with the problem of finding conditions that F = G*G a.e. where G is the strong boundary value function of a suitable operator valued analytic function defined in the disk \z\ < 1 or half-plane y>0. Mainly we are interested in special classes of functions in which such a factorization is always possible. Our study is motivated by the Fejér-Riesz theorem on the factorization of nonnegative trigonometric polynomials, and Ahiezers version [l ] of its generalization to entire functions of exponential type which are nonnegative on the real axis. Both results generalize to operator valued functions, and, in fact, both appear as special cases of a very general result (Theorem 3.1). More generally we present a unified treatment of the factorization problem, and thus much of §1 is expository. There we develop the theory of a corresponding abstract factorization problem for nonnegative Hubert space operators. Both the results and methods of §1 are purely operator theoretic. In §2 we show how the abstract theory relates to the theory of operator valued functions defined on the circle T or line R. The main applications to the factorization problem for nonnegative operator valued functions are deferred to §3. The factorization problem arises in the prediction theory of stationary stochastic processes. For this connection see Helson and Lowdenslager [ l l ] , Rozanov [27], and Wiener and Masani [28]. We wish to thank Professor Loren Pitt for calling our attention to the paper by E. Robinson [23]. We have extended Robinsons results in §2.

An operator-theoretic approach to theorems of the Pick-Nevanlinna and Loewner types. I

In this paper we formulate a general operator-theoretic result from which we deduce some old and some new function-theoretic interpolation theorems. The theorems we are concerned with have as prototype the Pick-Nevanlinna theorem and the Loewner theorem on restrictions of functions with positive imaginary part in the upper half plane.

Extension Theorems for Contraction Operators on Kreĭn Spaces

Notions of Julia and defect operators are used as a foundation for a theory of matrix extension and commutant lifting problems for contraction operators on Kreĭn spaces. The account includes a self-contained treatment of key propositions from the theory of Potapov, Ginsburg, Kreĭn, and Shmul’yan on the behavior of a contraction operator on negative subspaces. This theory is extended by an analysis of the behavior of the adjoint of a contraction operator on negative subspaces. Together, these results provide the technical input for the main extension theorems.

Restrictions of analytic functions. III

An operator theoretic method is used to characterize restrictions of boundary functions of H and H°° functions and to obtain a generalization of a theorem of Loewner.

On Generalized Schwarz-Pick Estimates

By Picks invariant form of Schwarzs lemma, an analytic function B ( z ) which is bounded by one in the unit disk D = { z : | z | at each point α of D . Recently, several authors [ 2, 10, 11 ] have obtained more general estimates for higher order derivatives. Best possible estimates are due to Ruscheweyh [ 12 ]. Below in §2 we use a Hilbert space method to derive Ruscheweyhs results. The operator method applies equally well to operator-valued functions, and this generalization is outlined in §3.

Schwarz-Pick Inequalities for the Schur-Agler Class on the Polydisk and Unit Ball

The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.

The Operator Fejer-Riesz Theorem

The Fejer-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrix- and operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblums operator generalization of the classical Fejer-Riesz theorem.

Ideals of square summable power series. II

The closed invariant subspaces of multiplication by z in H2 were determined by Beurling [1, Theorem IV, p. 253]. Vector generalizations of this theorem are known (Halmos [3] and the author [6]), but they involve an unnecessary use of analysis. We can now prove the theorem of [6] by purely algebraic and geometric methods. To emphasize these methods, we work with sequences, which we write as formal power series, rather than functions analytic in the unit disk. Let e be a Hilbert space with elements denoted by a, b, c, and with norm j * |. If b is a vector in C, then b is the linear functional on C such that ba = (a, b) for every a in C. A formal power series is a sequence (ao, a,, a2, * * ) written f(z) = Janzn with an indeterminate z. Let f(z) = 2anZn and g(z) = EbnZn be formal power series with coefficients an and bn in C; let B(z) = EBnzn be a formal power series whose coefficients Bn are (bounded) operators in C; let a be a complex number, and let c be a vector in C. Then f(z) +g(z), af(z), zf(z), and B(z)f(z) are the formal power series Z(an +bn)zn, (aan)Zn, (jan)zn, and ( X_0 Bkan-k)Zn, respectively. A sequence (fk(z)) of formal power series with coefficients in C is said to be formally convergent if, for each n = 0, 1, 2, * * * , the corresponding sequence of nth coefficients is convergent. Let C(z) be the Hilbert space of formal power series f(z) = anZn with coefficients an in C, such that

Recent Results and Unsolved Problems on Finite Convolution Operators

Finite convolution operators are studied by means of a complex Fourier transform technique. Questions concerning unicellularity and similarity are related to asymptotic properties of bounded analytic functions in a half-plane. The purpose of the paper is to survey recent work, and to call attention to open problems. The theory is illustrated by a list of examples.