Linear Operators, Part 2 |
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Page 866
... element the ideal xx is 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 ...
... element the ideal xx is 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 ...
Page 877
... element y in has an inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of Y is the same as its spectrum as an element of X. = e * = PROOF . If y1 exists as an element of Y then , since X and ...
... element y in has an inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of Y is the same as its spectrum as an element of X. = e * = PROOF . If y1 exists as an element of Y then , since X and ...
Page 1339
... elements is a { u ,, } - null element . Since a scalar multiple of a { u } -null element is evidently a { μ ,, } - null element , the family N ( { μ ,; } ) of { μ ,, } - null elements is a linear subspace of L2 ( { μ ,; } ) . We shall ...
... elements is a { u ,, } - null element . Since a scalar multiple of a { u } -null element is evidently a { μ ,, } - null element , the family N ( { μ ,; } ) of { μ ,, } - null elements is a linear subspace of L2 ( { μ ,; } ) . We shall ...
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 domain 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 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 tr(T transform unique unitary vanishes vector zero