Linear Operators: Spectral theory |
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Page 990
... closed linear subspace of L ( R ) which is determined by the char- acters in any neighborhood of its spectral set . Conversely , if q is in the L1 - closed linear manifold determined by the characters in some closed set F in R then o ...
... closed linear subspace of L ( R ) which is determined by the char- acters in any neighborhood of its spectral set . Conversely , if q is in the L1 - closed linear manifold determined by the characters in some closed set F in R then o ...
Page 1775
... 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 ...
... 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 ...
Page 1779
... linear manifold N in H if A is an orthonormal set contained in N and if x = N. x = Σ ( x , y ) y , νεα 12 THEOREM . Every closed linear manifold in contains an orthonormal basis for itself . PROOF . If the orthonormal sets in the closed ...
... linear manifold N in H if A is an orthonormal set contained in N and if x = N. x = Σ ( x , y ) y , νεα 12 THEOREM . Every closed linear manifold in contains an orthonormal basis for itself . PROOF . If the orthonormal sets in the closed ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero