Linear Operators: Spectral theory |
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Page 865
... contained in the boundary of o ( x ) . PROOF . Since the unit e in X is also in X , it follows that a regular element in X is regular in X. Thus po ( x ) p ( x ) or o ( x ) Co ( x ) . If λe bdy σ ( x ) , the boundary of σ ( x ) , then ...
... contained in the boundary of o ( x ) . PROOF . Since the unit e in X is also in X , it follows that a regular element in X is regular in X. Thus po ( x ) p ( x ) or o ( x ) Co ( x ) . If λe bdy σ ( x ) , the boundary of σ ( x ) , then ...
Page 866
... contained in a maximal ideal . Thus an element is contained in a maximal right ( left , two - sided ) ideal if and only if it has no inverse . The above facts about ideals are summarized in the following lemma . 12 LEMMA . The following ...
... contained in a maximal ideal . Thus an element is contained in a maximal right ( left , two - sided ) ideal if and only if it has no inverse . The above facts about ideals are summarized in the following lemma . 12 LEMMA . The following ...
Page 1162
... contained in a maximal ideal , but if an identity is not present this is not true and it becomes an important problem to find when a closed ideal is contained in a regular maximal ideal . Theorem 4.8 settles this question for L1 ( R ) ...
... contained in a maximal ideal , but if an identity is not present this is not true and it becomes an important problem to find when a closed ideal is contained in a regular maximal ideal . Theorem 4.8 settles this question for L1 ( R ) ...
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