Mathematical Sciences: Problems in Function--Theoretic      Operator Theory
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Operator theory is a central discipline in Modern Analysis.                Its origins lie in the study of mathematical physics and partial                differential equations in the early twentieth century.  Thus, it                was seen that numerous physical problems in the theory of                       equilibria, vibrations, quantum theory, etc. could be studied                   productively via the integral equations that model the phenomena.                                                                                               So it has been, that from the fertile minds of Hilbert, von                     Neumann, and other giants that the subject of operator theory has               grown to a central position in such investigations, and in core                 mathematics as well.  At the heart of this methodology is the                   deep investigation of the spectrum of an operator and the                       concommitant study of its invariant subspaces.  For the so-                     called self-adjoint operators, this theory is now a standard                    technique throughout analysis, and the spectral theorem provides                the necessary building blocks for all such operators.  Although                 not quite as complete, spectral theory is significantly well                    understood for an extensive generalization of the self-adjoint                  case; viz., the "normal" operators.  The current frontier,                      therefore, in the study of this structure theory rests in the                   non-normal theory.  A particularly important class of such                      operators are referred to as "subnormal".  They are significant                 for two reasons.  First, there is enough residual normal                        information to attempt a deeper understanding for their inherent                structure.  Recent results, such as the fact that subnornmal                    operators are reflexive and have invariant subspaces, have given                an added impetus to the study of their spectral properties.                     Secondly, since many explicit operators that arise in complex                   function theory, differential geometry, and approximation theory                are subnormal, there is a broad range of applicability.                              Professor Kriete is a leader in the spectral theory of                     subnormal operators and its relationship to classical analysis,                 especially polynomial approximation.  His contributions date back               to a landmark paper on the spectral theory of the Cesaro                        operator in 1972.  This work created new directions in spectral                 theory, with implications for the theory of special functions.                  Professor Kriete's work has had a particularly strong influence                 on the Russian school of operator theory.  Professor Kriete's                   work of the past few years has been of the same high calibre as                 his earlier work.  In the present proposal, Professor Kriete will               investigate several problems at the interface of operator theory                and function theory, all of which exhibit both significanct                     interest and difficulty.  One such problem is the determination                 of when a composition operator on classical spaces in complex                   function theory has a subnormal adjoint.  A second involves                     functional models for contraction operators.  He will also                      continue his applications to classical polynomial approximation                 with emphasis on questions of splitting.
{'FirstName': 'Thomas', 'LastName': 'Kriete', 'PI_MID_INIT': 'L', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Thomas L Kriete', 'EmailAddress': 'tlk8q@virginia.edu', 'NSF_ID': '000244719', 'StartDate': '06/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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