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An invariant for certain operator algebras.

R W Carey1, J D Pincus

  • 1Department of Mathematics, State University of New York, Stony Brook, N.Y. 11790.

Proceedings of the National Academy of Sciences of the United States of America
|May 1, 1974
PubMed
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The principal function invariant, associated with operators, remains stable under trace class perturbations. This function extends operator index calculations and aids in understanding invariant subspaces and point spectra.

Area of Science:

  • Operator theory
  • Functional analysis
  • Algebraic topology

Background:

  • The principal function is a key invariant for operators with trace-class self-commutators.
  • Understanding operator properties requires robust invariants stable under perturbations.

Purpose of the Study:

  • To demonstrate the invariance of the principal function under trace-class perturbations.
  • To explore the principal function's connection to the maximal ideal space of C*-algebras.
  • To investigate the principal function's role in invariant subspaces and point spectra.

Main Methods:

  • Unitary invariance is used to associate the principal function with operators.
  • Trace-class perturbations are applied to operators to test invariance.
  • Connections are explored with the maximal ideal space of the C*-algebra generated by the operator.

Related Experiment Videos

  • The principal function's behavior with non-integer values is analyzed.

    • Main Results:

    • The principal function is invariant under trace-class perturbations of the operator.
    • It serves as an extension of the operator index (T-z) across the entire complex plane.
    • The principal function relates to the maximal ideal space of the generated C*-algebra under specific conditions.
    • The principal function, even with non-integer values, is crucial for invariant subspace existence and point spectrum determination.

    Conclusions:

    • The principal function is a powerful invariant for operators, stable under perturbations.
    • It offers deeper insights into operator structure, including spectral properties and ideal spaces.
    • Its non-integer values highlight its broad applicability in operator theory.