Stephan Ramon Garcia

Some New Classes of Complex Symmetric Operators

We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).

Complex symmetric partial isometries

An operator T∈B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C:H→H so that T=CT∗C. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension ⩽4 is complex symmetric.

Recent Progress on Truncated Toeplitz Operators

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason’s seminal paper (Oper. Matrices 1(4):491–526, 2007).

Mathematical and physical aspects of complex symmetric operators

Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-Hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics and complex variables.

Which Weighted Composition Operators are Complex Symmetric

Recent work by several authors has revealed the existence of many unexpected classes of normal weighted composition operators. On the other hand, it is known that every normal operator is a complex symmetric operator. We therefore undertake the study of complex symmetric weighted composition operators, identifying several new classes of such operators.

Norm estimates of complex symmetric operators applied to quantum systems

This paper communicates recent results in the theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schrodinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schrodinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent norm for Schrodinger operators appearing in the complex scaling theory of resonances.

A Non-Linear Extremal Problem on the Hardy Space

We relate some classical extremal problems on the Hardy space to norms of truncated Toeplitz operators and complex symmetric operators.

Approximate Antilinear Eigenvalue Problems and Related Inequalities

If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum σ(|T|) of √T*T can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator T: H → H), in terms of the spectra of the selfadjoint operators ReT and ImT, restricting the possible location of elements of σ(|T|). A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.

The graphic nature of Gaussian periods

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework within which to study certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, our approach reveals that these classical objects display dazzling visual patterns of great complexity and remarkable subtlety.

The Eigenstructure of Complex Symmetric Operators

We discuss several algebraic and analytic aspects of the eigenstructure (si.e., eigenvalues, eigenvectors, and generalized eigenvectors) of complex symmetric operators. In particular, we examine the relationship between the bilinear form [x,y] = induced by a conjugation C on a complex Hilbert space H and the eigenstructure of a bounded linear operator T: H → H which is C-symmetric (T = CT*C).