Linear Operators, Part 2 |
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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 1687
... closure of D , it follows that ( UD ) must consist of one or another of the hemispheres V1 = { xe V x1 > 0 } or V_ ... closure in E " of whose support is disjoint from the curved bound- ary of V , and all of whose derivatives of order at ...
... closure of D , it follows that ( UD ) must consist of one or another of the hemispheres V1 = { xe V x1 > 0 } or V_ ... closure in E " of whose support is disjoint from the curved bound- ary of V , and all of whose derivatives of order at ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero