Linear Operators: Spectral theory |
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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 = Þ × X where ...
... 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 = Þ × X where ...
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 ( L1 ( 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 ( L1 ( R ) ) . Its elements have ...
Page 1493
... unit circle under the map 1 includes a set of m analytic arcs all intersecting at equal angles / m at the point 2. If m > 1 , not all of these arcs can possibly lie along the real axis . Hence we must have m = 1 , so that 1 is one - to ...
... unit circle under the map 1 includes a set of m analytic arcs all intersecting at equal angles / m at the point 2. If m > 1 , not all of these arcs can possibly lie along the real axis . Hence we must have m = 1 , so that 1 is one - to ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero