Linear Operators: Spectral theory |
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Page 1100
... prove ( b ) in general , we have only to prove ( b ) in the special case in which A and B have finite - dimensional ranges . But then the inequality in ( b ) is plainly a special case of ( a ) , while the identity ( 3 ) follows readily ...
... prove ( b ) in general , we have only to prove ( b ) in the special case in which A and B have finite - dimensional ranges . But then the inequality in ( b ) is plainly a special case of ( a ) , while the identity ( 3 ) follows readily ...
Page 1393
... prove that TX is closed if TY is closed , we shall prove more generally that the sum of a closed subspace 3 of a B - space , and of a finite dimensional space Î , is closed . It is clear that proceed- ing inductively we may assume ...
... prove that TX is closed if TY is closed , we shall prove more generally that the sum of a closed subspace 3 of a B - space , and of a finite dimensional space Î , is closed . It is clear that proceed- ing inductively we may assume ...
Page 1563
... Prove that = 0 ( √ ( bn ― an ) ) . ( b ) Prove that the essential spectrum of 7 contains the positive semi - axis . ( Hint : Apply Theorem 7.1 . ) G41 Suppose that the function q is bounded below . Suppose that the origin belongs to ...
... Prove that = 0 ( √ ( bn ― an ) ) . ( b ) Prove that the essential spectrum of 7 contains the positive semi - axis . ( Hint : Apply Theorem 7.1 . ) G41 Suppose that the function q is bounded below . Suppose that the origin belongs to ...
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