Linear Operators: Spectral theory |
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Page 978
... Closure Theorems As in the preceding section the letter R will stand for a non- discrete locally compact Abelian group and integration will always be performed with respect to a Haar measure on the group . It was observed in Corollary ...
... Closure Theorems As in the preceding section the letter R will stand for a non- discrete locally compact Abelian group and integration will always be performed with respect to a Haar measure on the group . It was observed in Corollary ...
Page 1226
... closure , and written T. 8 LEMMA . ( a ) The closure T of T is the restriction of T * to the closure of D ( T ) in the Hilbert space D ( T * ) . ( b ) The operator T and its closure have the same closed extensions . ( c ) The operator T ...
... closure , and written T. 8 LEMMA . ( a ) The closure T of T is the restriction of T * to the closure of D ( T ) in the Hilbert space D ( T * ) . ( b ) The operator T and its closure have the same closed extensions . ( c ) The operator T ...
Page 1686
... closure of D. = 1 PROOF . Cover the closure of D with a finite collection of bounded open sets U each of which is either disjoint from the boundary of D or is differentiably equivalent to a spherical neighborhood V of the origin of E ...
... closure of D. = 1 PROOF . Cover the closure of D with a finite collection of bounded open sets U each of which is either disjoint from the boundary of D or is differentiably equivalent to a spherical neighborhood V of the origin of E ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero