Linear Operators, Part 2 |
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Page 1272
... indices are d = 0 , 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 ...
... indices are d = 0 , 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 ...
Page 1398
... indices of To ( t ) is k , then for λ o , ( t ) the equation to ho has at least k linearly independent solutions in L2 ( I ) . e λσ PROOF . By Theorem 2.10 and XII.4.7 ( c ) , the adjoint of T1 ( ) is T1 ( 7 ) . The desired result thus ...
... indices of To ( t ) is k , then for λ o , ( t ) the equation to ho has at least k linearly independent solutions in L2 ( I ) . e λσ PROOF . By Theorem 2.10 and XII.4.7 ( c ) , the adjoint of T1 ( ) is T1 ( 7 ) . The desired result thus ...
Page 1454
... indices of T are equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) -K ( x , x ) , x = D ( T ) . The proof of Theorem 5 now shows that σ ( T ) is a subset of the half- axis ...
... indices of T are equal . PROOF . To prove ( a ) , note that if T is bounded below , there exists a constant K such that ( Tx , x ) -K ( x , x ) , x = D ( T ) . The proof of Theorem 5 now shows that σ ( T ) is a subset of the half- axis ...
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BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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(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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero