Linear Operators, Part 2 |
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Page 1272
... deficiency indices are different from zero . A maximal symmetric operator is one which has no proper symmetric extensions ; hence , a closed symmetric operator is maximal if at least one of its deficiency indices is zero . If both are ...
... deficiency indices are different from zero . A maximal symmetric operator is one which has no proper symmetric extensions ; hence , a closed symmetric operator is maximal if at least one of its deficiency indices is zero . If both are ...
Page 1398
... deficiency indices of To ( T ) is k , then for λo , ( T ) the equation to λo has at least k linearly independent solutions in L¿ ( I ) . = λσ PROOF . By Theorem 2.10 and XII.4.7 ( c ) , the adjoint of To ( t ) is T1 ( 7 ) . The desired ...
... deficiency indices of To ( T ) is k , then for λo , ( T ) the equation to λo has at least k linearly independent solutions in L¿ ( I ) . = λσ PROOF . By Theorem 2.10 and XII.4.7 ( c ) , the adjoint of To ( t ) is T1 ( 7 ) . The desired ...
Page 1611
... deficiency indices of t are equal ( 6.6 ) . ( 2 ) In particular , the deficiency indices are equal if 7 is bounded below . ( 3 ) If for some real or complex λ all solutions of the equation ( λ − t ) ƒ = 0 are square - integrable , then ...
... deficiency indices of t are equal ( 6.6 ) . ( 2 ) In particular , the deficiency indices are equal if 7 is bounded below . ( 3 ) If for some real or complex λ all solutions of the equation ( λ − t ) ƒ = 0 are square - integrable , then ...
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