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The complex pairing on equivalence classes
Definition
For , the proposed pairing is
The integral is computed from measurable representatives. By Complex Holder, Minkowski, and the quotient norm, and , so this representative expression is defined. The measurable and integration conventions are those of Complex Lp classes and Euclidean test-function conventions. This fixes the first-variable-linear convention. The following theorem, named in justified_by, establishes class invariance and the inner-product axioms; the present definition does not assume that obligation.
Depends on
Used by
- Non-invertible elliptic shifts form a discrete set in the self-adjoint case Corollary
- E-valued Dolbeault operators and their L2 structure Definition
- Hermitian metric and L² pairing on a compact Riemann surface Definition
- Left and right regular unitary representations of an LCH group Definition
- The Bergman space A²(Ω) and the Bergman kernel Definition
- The Hardy boundary space, Szegő projection and Szegő kernel on a smoothly bounded domain Definition
- The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface Definition
- Weighted L2 spaces and maximal dbar operators Definition
- Ball monomial norms, Bergman and Szegő kernels of the ball Example
- Convolution of matrix coefficients on a compact group Example
- The resolvent norm blows up at an eigenvalue Example
- Complex completeness, density, and inner product: the consumer interface Lemma
- Polynomial traces, monomial basis and bounded evaluation for the ball Hardy space Lemma
- Restriction and extension estimates are dual Lemma
- Scalar unitisation of L¹ of an LCA group: characters, spectrum and identity criterion Lemma
- Smoothness of the Bergman kernel and positivity of its diagonal on bounded domains Lemma
- The elliptic resolvent identity Lemma
- The weighted discrete-series space is a Hilbert space with K-type basis Lemma
- TT-star reduces extension to convolution with the surface-measure transform Lemma
- Harmonic star duality for line bundle valued dolbeault cohomology Theorem
- The complex L² pairing is well-defined and satisfies Cauchy–Schwarz Theorem
- The Hulanicki–Reiter weak containment criterion for amenability Theorem
- The symmetric elliptic form operator is self-adjoint with compact resolvent Theorem
Dependency tree · two levels
26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)