Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 868
Nelson Dunford, Jacob T. Schwartz. 2. Commutative B - Algebras In case is a commutative B - algebra every ideal is two - sided and the quotient algebra X / is again a commutative algebra . It will be a B - algebra if is closed ( 1.13 ) ...
Nelson Dunford, Jacob T. Schwartz. 2. Commutative B - Algebras In case is a commutative B - algebra every ideal is two - sided and the quotient algebra X / is again a commutative algebra . It will be a B - algebra if is closed ( 1.13 ) ...
Page 869
... commutative B - algebra into the complex number system is continuous . 4 LEMMA . Let M be the set of maximal ideals in the commutative B - algebra X. Then x ( M ) = o ( x ) and sup ( M ) = lim " / " . MEM n PROOF . Since the element x ...
... commutative B - algebra into the complex number system is continuous . 4 LEMMA . Let M be the set of maximal ideals in the commutative B - algebra X. Then x ( M ) = o ( x ) and sup ( M ) = lim " / " . MEM n PROOF . Since the element x ...
Page 882
... commutative Banach algebra . 14 If ƒ is in L1 ( —∞ , ∞ ) , and if λ ( E ) = √g f ( s ) ds show that f ∞ ( 2 * μ ) ... ( commutative ) B - algebra has only trivial ideals , then it is isometrically isomorphic to the complex numbers . Show ...
... commutative Banach algebra . 14 If ƒ is in L1 ( —∞ , ∞ ) , and if λ ( E ) = √g f ( s ) ds show that f ∞ ( 2 * μ ) ... ( commutative ) B - algebra has only trivial ideals , then it is isometrically isomorphic to the complex numbers . Show ...
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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF 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