Linear Operators: Spectral theory |
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Page 980
... complex valued homomorphism H on either A or A1 is continuous and has H ( I ) = 1 it follows that H is completely determined by the values it takes on elements of the form T ( f ) with f in L1 ( R ) . Thus each of the spaces M and M1 ...
... complex valued homomorphism H on either A or A1 is continuous and has H ( I ) = 1 it follows that H is completely determined by the values it takes on elements of the form T ( f ) with f in L1 ( R ) . Thus each of the spaces M and M1 ...
Page 1036
... complex valued function on the B - space of all Hilbert - Schmidt operators . ( * ) PROOF . First note that if 5 is a complex number with || < 1 then log e $ ( 1—5 ) = 5− ( 5+ 3/2 52 + 3 / 53 + . ) · = O ( 152 ) , as 0. Let f ( 5 ) ...
... complex valued function on the B - space of all Hilbert - Schmidt operators . ( * ) PROOF . First note that if 5 is a complex number with || < 1 then log e $ ( 1—5 ) = 5− ( 5+ 3/2 52 + 3 / 53 + . ) · = O ( 152 ) , as 0. Let f ( 5 ) ...
Page 1281
... complex - valued function integrable over every compact subinterval of I. Let to e I , and let co , C1 , ... , C - 1 be an arbitrary set of n complex numbers . Then there exists a unique fe A " ( I ) such that ( a ) rf = g , i ( b ) ( a ) ...
... complex - valued function integrable over every compact subinterval of I. Let to e I , and let co , C1 , ... , C - 1 be an arbitrary set of n complex numbers . Then there exists a unique fe A " ( I ) such that ( a ) rf = g , i ( b ) ( a ) ...
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