Composition satisfies |ST|\le|S|,|T|

Depends on

• The operator norm as the least bound and as the unit-sphere or unit-ball supremum
• The spaces \(\mathcal B(X,Y)\) and \(\mathcal B(X)\) of bounded linear operators

Used by

• A semigroup with unbounded generator is not norm continuous at zero Corollary
• Contraction Hille-Yosida theorem Corollary
• C k map between Banach spaces Definition
• The transpose of a bounded operator Definition
• Adjoint, norm and trace of an operator of rank at most one Example
• The square of the Volterra operator has zero trace Example
• A semigroup with continuity at zero is uniformly bounded on every compact time interval Lemma
• A separably acting abelian von Neumann algebra has a self-adjoint generator Lemma
• Compositions with a compact operator are compact Lemma
• Continuity at time zero implies continuity of every orbit Lemma
• Neumann series and small perturbations of bounded inverses Lemma
• Resolvent identity and holomorphy for a closed operator Lemma
• Spectral product from traces of powers Lemma
• The exponential series of a bounded operator Lemma
• The generator commutes with the semigroup on its domain Lemma
• Bounded Yosida semigroups converge to the generated semigroup Theorem
• Chain sum product and composition rules for Banach derivatives Theorem
• Exponential bound for a C0-semigroup Theorem
• Hilbert Schmidt operators form a two sided ideal Theorem
• Hilbert-adjoint identities Theorem
• Inverse function theorem for Banach spaces Theorem
• Lumer-Phillips generation theorem Theorem
• Measurable essentially bounded operator fields act decomposably Theorem
• Pvm integral is a star homomorphism Theorem
• The method of continuity for a uniformly estimated affine family of bounded operators Theorem
• Trace class is a two sided Banach operator ideal Theorem

Dependency tree · two levels

5 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)
• Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)