Linear Operators: Spectral theory |
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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 L2 ( I ) . PROOF . By Theorem 2.10 and XII.4.7 ( c ) , the adjoint of Tô ( 7 ) 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 L2 ( I ) . PROOF . By Theorem 2.10 and XII.4.7 ( c ) , the adjoint of Tô ( 7 ) is T1 ( 7 ) . The desired ...
Page 1611
... deficiency indices of t are equal ( 6.6 ) . τ ( 2 ) In particular , the deficiency indices are equal if 7 is bounded below . ( 3 ) If for some real or complex 2 all solutions of the equation ( 1 - T ) = 0 are square - integrable , then ...
... deficiency indices of t are equal ( 6.6 ) . τ ( 2 ) In particular , the deficiency indices are equal if 7 is bounded below . ( 3 ) If for some real or complex 2 all solutions of the equation ( 1 - T ) = 0 are square - integrable , then ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero