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 993
... closure . Then it follows from what has just been demonstrated that av , = ayu , ay , i.e. , ap is independent of ... closure we have ( i ) τφτίχν = χν · Since every compact set in R is contained in an open set with compact closure it ...
... closure . Then it follows from what has just been demonstrated that av , = ayu , ay , i.e. , ap is independent of ... closure we have ( i ) τφτίχν = χν · Since every compact set in R is contained in an open set with compact closure it ...
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 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero