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 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero