Linear Operators: Spectral theory |
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Page 1775
... manifold M in is a closed linear manifold complementary to M , i.e. , H = M → N. PROOF . It follows from the linearity and the continuity of the scalar product ( Theorem 1 ) that the orthocomplement of any set M is a closed linear ...
... manifold M in is a closed linear manifold complementary to M , i.e. , H = M → N. PROOF . It follows from the linearity and the continuity of the scalar product ( Theorem 1 ) that the orthocomplement of any set M is a closed linear ...
Page 1779
... manifold in 5 contains an orthonormal basis for itself . PROOF . If the orthonormal sets in the closed linear manifold M are ordered by inclusion , it is seen from Zorn's lemma ( I.2.7 ) that there is a maximal one A which determines ...
... manifold in 5 contains an orthonormal basis for itself . PROOF . If the orthonormal sets in the closed linear manifold M are ordered by inclusion , it is seen from Zorn's lemma ( I.2.7 ) that there is a maximal one A which determines ...
Page 1912
... Manifold , closed linear , spanned by a set , II.1.4 ( 50 ) in a linear space , ( 36 ) . ( See also Linear manifold ) orthogonal , in Hilbert space , IV.4.3 ( 249 ) Mapping . ( See also Function ) interior principle , II.2.1 ( 55 ) ...
... Manifold , closed linear , spanned by a set , II.1.4 ( 50 ) in a linear space , ( 36 ) . ( See also Linear manifold ) orthogonal , in Hilbert space , IV.4.3 ( 249 ) Mapping . ( See also Function ) interior principle , II.2.1 ( 55 ) ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero