Linear Operators, Part 2 |
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Page 860
... unit e . We shall show how a unit may be adjoined to such an algebra so that the extended algebra is a B - algebra . Let X be an alge- bra satisfying all the requirements of a B - algebra except that X has no unit . Let X1 = XX where is ...
... unit e . We shall show how a unit may be adjoined to such an algebra so that the extended algebra is a B - algebra . Let X be an alge- bra satisfying all the requirements of a B - algebra except that X has no unit . Let X1 = XX where is ...
Page 865
... unit as X. 9 LEMMA . Let x be an element of a B - subalgebra Xo of X whose unit is the same as that of X. Then o ( x ) С σ ( x ) while the boundary of σ ( x ) is contained in the boundary of o ( x ) . PROOF . Since the unit e in X is ...
... unit as X. 9 LEMMA . Let x be an element of a B - subalgebra Xo of X whose unit is the same as that of X. Then o ( x ) С σ ( x ) while the boundary of σ ( x ) is contained in the boundary of o ( x ) . PROOF . Since the unit e in X is ...
Page 979
... unit I in B ( L2 ( R ) ) nor does its closure T ( L1 ( R ) ) in the uniform operator topology contain the unit . The algebra A is , by definition , the B - algebra obtained by adjoining the unit I to T ( LÎ ( R ) ) . Its elements have ...
... unit I in B ( L2 ( R ) ) nor does its closure T ( L1 ( R ) ) in the uniform operator topology contain the unit . The algebra A is , by definition , the B - algebra obtained by adjoining the unit I to T ( LÎ ( R ) ) . Its elements have ...
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