Abstract
The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite-dimensional subspaces. As a consequence, given a compact manifold M endowed with a positive measure, we introduce a notion of the operator’s full symbol adapted to the Fourier analysis relative to a fixed elliptic operator E. We give a description of Fourier multipliers, or of operators invariant relative to E. We apply these concepts to study Schatten classes of operators on L2(M) and to obtain a formula for the trace of trace class operators. We also apply it to provide conditions for operators between Lp-spaces to be r-nuclear in the sense of Grothendieck.
Highlights
Let M be a closed manifold of dimension n endowed with a positive measure dx
We show this by relating the symbols introduced in this paper to matrix-valued symbols on compact Lie groups developed in [RT13] and in [RT10]
In Theorem 2.1 below, we discuss the abstract notion of symbol for operators densely defined in a general Hilbert space H and give several alternative formulations for invariant operators, or for Fourier multipliers, relative to a fixed partition of H into a direct sum H = j H j of finite-dimensional subspaces
Summary
Let M be a closed manifold (i.e., a compact smooth manifold without boundary) of dimension n endowed with a positive measure dx. In the case of compact Lie groups, our results extend results on Schatten classes and on r-nuclear operators on Lp spaces that have been obtained in [DR13] and [DR14b]. We show this by relating the symbols introduced in this paper to matrix-valued symbols on compact Lie groups developed in [RT13] and in [RT10]. In Theorem 2.1 below, we discuss the abstract notion of symbol for operators densely defined in a general Hilbert space H and give several alternative formulations for invariant operators, or for Fourier multipliers, relative to a fixed partition of H into a direct sum H = j H j of finite-dimensional subspaces.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have