Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1224
... symmetric then every symmetric extension T1⁄2 of T1 , and , in particular , every self adjoint extension of T1 , satisfies T1 CT , CT C T * . 1 2 2 2 PROOF . If T , CT , and ye D ( T ) , then ( x , T * y ) = ( T2x , y ) = ( T1x , y ) ...
... symmetric then every symmetric extension T1⁄2 of T1 , and , in particular , every self adjoint extension of T1 , satisfies T1 CT , CT C T * . 1 2 2 2 PROOF . If T , CT , and ye D ( T ) , then ( x , T * y ) = ( T2x , y ) = ( T1x , y ) ...
Page 1236
... symmetric extension of T is the restriction of T * to the subspace of D ( T * ) determined by a symmetric family of boundary conditions , B ( x ) = 0 , i = 1 , ... , k . Conversely , every such restriction T1 of T * is a closed ...
... symmetric extension of T is the restriction of T * to the subspace of D ( T * ) determined by a symmetric family of boundary conditions , B ( x ) = 0 , i = 1 , ... , k . Conversely , every such restriction T1 of T * is a closed ...
Page 1272
... symmetric operators . If T is a symmetric operator with dense domain , then it has proper symmetric extensions provided both of its deficiency indices are different from zero . A maximal symmetric operator is one which has no proper ...
... symmetric operators . If T is a symmetric operator with dense domain , then it has proper symmetric extensions provided both of its deficiency indices are different from zero . A maximal symmetric operator is one which has no proper ...
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