Linear Operators: Spectral theory |
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Page 1272
... indices are d o , d_ 1 . The operator T1 is called an elementary symmetric operator . It may be proved that if T is maximal symmetric with indices d = 0 , d_ = n ( where n is any cardinal number ) , then may be broken into a direct sum ...
... indices are d o , d_ 1 . The operator T1 is called an elementary symmetric operator . It may be proved that if T is maximal symmetric with indices d = 0 , d_ = n ( where n is any cardinal number ) , then may be broken into a direct sum ...
Page 1398
... indices of To ( t ) is k , then for 2 ‡ σ , ( 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 T。( t ) is T1 ( 7 ) . The desired result thus ...
... indices of To ( t ) is k , then for 2 ‡ σ , ( 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 T。( t ) is T1 ( 7 ) . The desired result thus ...
Page 1611
... indices of 7 are equal ( 6.6 ) . ( 2 ) In particular , the deficiency indices are equal if t is bounded below . ( 3 ) If for some real or complex 2 all solutions of the equation ( 2-7 ) = 0 are square - integrable , then the deficiency ...
... indices of 7 are equal ( 6.6 ) . ( 2 ) In particular , the deficiency indices are equal if t is bounded below . ( 3 ) If for some real or complex 2 all solutions of the equation ( 2-7 ) = 0 are square - integrable , then the deficiency ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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