Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 81
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. 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 " as ...
... closure of D. 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 " as ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
59 other sections not shown
Common terms and phrases
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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero