Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos's Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse's Theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two analytic functions), whose co-analytic part has a "coprime decomposition," is normal or analytic. We also prove that the coprime decomposition condition is essential. Finally, we examine a well known conjecture, of whether every submormal Toeplitz operator with finite rank self-commutator is normal or analytic.Comment: Final version, accepted for publication in Journal of Functional Analysi

Hyponormality and Subnormality of Block Toeplitz Operators

In this paper we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space $H^2_{\mathbb{C}^n}$ of the unit circle. Firstly, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator. Secondly, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if $\Phi$ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator $T_{\Phi}$ whose square is also hyponormal must be either normal or analytic. Thirdly, using the subnormal theory of block Toeplitz operators, we give an answer to the following "Toeplitz completion" problem: Find the unspecified Toeplitz entries of the partial block Toeplitz matrix A:=[U^*& ? ?&U^*] so that $A$ becomes subnormal, where $U$ is the unilateral shift on $H^2$.Comment: Final version, accepted for publication in Advances in Mathematic

A gap between hyponormality and subnormality for block Toeplitz operators

AbstractThis paper concerns a gap between hyponormality and subnormality for block Toeplitz operators. We show that there is no gap between 2-hyponormality and subnormality for a certain class of trigonometric block Toeplitz operators (e.g., its co-analytic outer coefficient is invertible). In addition we consider the extremal cases for the hyponormality of trigonometric block Toeplitz operators: in this case, hyponormality and normality coincide

Comparison of the Plasma Metabolome Profiles Between the Internal Thoracic Artery and Ascending Aorta in Patients Undergoing Coronary Artery Bypass Graft Surgery Using Gas Chromatography Time-of-Flight Mass Spectrometry.

BackgroundThe left internal thoracic artery (LITA) has been used as the first conduit of choice in coronary artery bypass grafting (CABG) because of excellent long-term patency and outcomes. However, no studies have examined substances other than nitric oxide that could be beneficial for the bypass conduit, native coronary artery or ischemic myocardium. This study was conducted to evaluate differences in metabolic profiles between the LITA and ascending aorta using gas chromatography-time of flight-mass spectrometry (GC-TOF-MS).MethodsTwenty patients who underwent CABG using the LITA were prospectively enrolled. Plasma samples were collected simultaneously from the LITA and ascending aorta. GC-TOF-MS based untargeted metabolomic analyses were performed and a 2-step volcano plot analysis was used to identify distinguishable markers from two plasma metabolome profiles. Semi-quantitative and quantitative analyses were performed using GC-TOF-MS and enzyme-linked immunosorbent assay, respectively, after selecting target metabolites based on the metabolite set enrichment analysis.ResultsInitial volcano plot analysis demonstrated 5 possible markers among 851 peaks detected. The final analysis demonstrated that the L-cysteine peak was significantly higher in the LITA than in the ascending aorta (fold change = 1.86). The concentrations of intermediate metabolites such as L-cysteine, L-methionine and L-cystine in the 'cysteine and methionine metabolism pathway' were significantly higher in the LITA than in the ascending aorta (2.0-, 1.4- and 1.2-fold, respectively). Quantitative analysis showed that the concentration of hydrogen sulfide (H₂S) was significantly higher in the LITA.ConclusionThe plasma metabolome profiles of the LITA and ascending aorta were different, particularly higher plasma concentrations of L-cysteine and H₂S in the LITA

Orthogonality properties of transverse eigenmodes of phase conjugate optical resonators

The orthogonality properties of the transverse eigenmodes of optical resonators which have phase conjugate mirrors at both ends are derived. As in conventional resonators and also resonators with only one phase conjugate mirror, it is shown that the transverse eigenmodes are essentially biorthogonal, a relation which is satisfied between the set of modes propagating in one direction around the resonator and the adjoint set of modes propagating in the reverse direction