A truncated Toeplitz operator is the compression of a Toeplitz operator on the Hardy space H2 to a model subspace of H2. Such an operator has a family of symbols, and a basic goal is to understand the interplay between the operator, qua operator, and its symbols. This paper studies the question for self-adjoint truncated Toeplitz operators on finite-dimensional model spaces. The main focus is the two-dimensional case.

We study a minimum problem and associated maximum problem for finite, complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size $(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on ${\mathcal P}_N$, the space of complex polynomials of degrees at most $N$, with the usual Hilbert space structure it inherits as a subspace of $L^2$ of the unit circle. The operator $A$ is the compression to ${\mathcal P}_N$ of the multiplication operator on $L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of indices between $-N$ and $N$ match the matrix entries of $A$. Our minimum problem is to minimize the $L^{\infty}$ norm of such inducers. We show there is a unique one of minimum norm, and we describe it. The associated maximum problem asks for the maximum of the ratio of the preceding minimum to the operator norm. That problem remains largely open. We present some suggestive numerical evidence.

This article is the story of how the author had the good fortune to be able to prove the primordial version of the commutant lifting theorem. The phrase "good fortune" is used advisedly. The story begins with the intersection of two lives, Paul's and the author's.

Henry Helson, a leading figure in harmonic analysis whose ideas have had an enormous impact, died on January 10, 2010. Born on June 2, 1927, Henry grew up in Bryn Mawr, where his father Harry was a professor, carrying out the research that would make him one of the eminent psychologists of his era. His mother Lida, despite her upbringing as a fundamentalist Lutheran, found her way to the Quaker faith when Henry and his sister Martha were children. All three of them became and remained committed Quakers. Henry’s exceptional mathematical abilities shone through early. Although his father Harry hoped Henry would go into physics, Henry was hooked on mathematics. He entered Harvard as an undergraduate, graduating in 1947. His education had been interrupted for thirteen months by military service, which for him, as a Quaker, was a loathsome experience. On the eve of his graduation Henry was awarded a Harvard traveling fellowship. The fellowship enabled him to fulfill an intense desire he had nurtured to visit Europe. In the academic year 1947–1948 he visited London, Paris, Prague, and Vienna, but he spent most of the year in Poland, first in Warsaw, then in Wroclaw. Why Poland? The circumstances are explained in Henry’s essay [5]. He writes: “. . . [I] was crazy to see the destruction caused by the war in Europe. . . . The most desperate place in Europe seemed to be Poland, and there were mathematicians in Poland.” Another passage from the same essay reads:

It is shown that, in a proper coinvariant subspace of the shift operator on the Hardy space H(2), a densely defined operator that commutes with the commutant of the restricted backward shift is closable. A connection between this result and a case of the transitive algebra problem is discussed.

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H(2), with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H(2) are also studied.

A complex-valued function on the nonnegative real axis is said to be slowly oscillating if it is continuous, bounded, and differs from each of its translates by a function that vanishes at infinity. The family of such functions forms a commutative C*-algebra under the supremum norm. This paper investigates the topology of the Gelfand space of that algebra

Complex numbers Complex differentiation Linear-fractional transformations Elementary functions Power series Complex integration Core versions of Cauchy's theorem, and consequences Laurent series and isolated singularities Cauchy's theorem Further development of basic complex function theory Appendix 1: Sufficient condition for differentiability Appendix 2: Two instances of the chain rule Appendix 3: Groups, and linear-fractional transformations Appendix 4: Differentiation under the integral sign References Index.

Compressions of Toeplitz operators to coinvariant subspaces of H(2) are studied. Several characerizations of such operators are obtained; in particular, those of rank one are described. The paper is partly expository. Open questions are raised.

This is a survey of the many roles Aleksandrov-Clark measures play in questions of function theory and operator theory.

A product representation is obtained for outer func- tions in the unit disk whose boundary functions are real valued. Our treatment is based on work of A.B. Aleksandrov and A.G. Poltoratski.

482 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Thomas H. Wolff, a leading analyst and a winner of the Salem and Bôcher Prizes, was killed in an automobile accident on July 31, 2000, when he was forty-six years old. Tom was raised in a mathematical environment. His uncle, Clifford Gardner, was a professor at NYU’s Courant Institute of Mathematics for many years, and Tom’s mother, Lucile, was a technical editor of volume 1 of the English translation of the celebrated book Methods of Mathematical Physics by Courant and Hilbert. Tom was an undergraduate at Harvard, where, he once told me, he regularly played poker with a fellow student named Bill Gates. After graduating from Harvard in 1975, Tom went to Berkeley, where he got his Ph.D. under Don Sarason in 1979. Tom then spent one year at the University of Washington and two at the University of Chicago before coming to Caltech in the fall of 1982 as an assistant professor. Tom spent most of the rest of his career at Caltech, although, for personal reasons, he resigned twice, spending two years (1986–88) at Courant and three (1992–95) at Berkeley. His promotion or appointment to a professorship at Caltech three times is a record for our institution. Tom is survived by his widow, Carol Shubin, a mathematics professor at California State University, Northridge; two sons (aged three and five at his death); his parents; and two sisters. —Barry Simon Lennart Carleson

The space D(μ) associated with a positive measure μ on the unit circle is a Hilbert space made from the holomorphic functions in the unit disk whose derivatives are square integrable when weighted against the Poisson integral of μ. In this paper the structure of D(μ) is investigated for the case where μ is a finite sum of atoms. The wandering vectors of the shift operator on D(μ) are described.

Given a nonextreme point bof the unit ball of H ∞, the multipliers of the de Branges-Rovnyak space H(b) lie in an auxiliary space M (ā) ∩ H ∞, where a is a function in H ∞ that is associated with b and M (ā) is the range of the Toeplitz operator T ā on H 2. An example is constructed here to show that M (ā) ∩ H ∞ need not be an algebra. This contrasts with the case wherebis an extreme point, where the analogous auxiliary space is always an algebra.