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 L ( R ) . Thus each of the spaces M and M1 has ...
... 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 L ( R ) . Thus each of the spaces M and M1 has ...
Page 1036
... complex valued function on the B - space of all Hilbert - Schmidt operators . ( * ) PROOF . First note that if is a complex number with || < 1 then log e $ ( 1—5 ) = 5—15 + 1 / 252 + 31 53+ . ) .. = 0 ( 52 ) , as 5 → 0. Let ƒ ( 5 ) = 5 ...
... complex valued function on the B - space of all Hilbert - Schmidt operators . ( * ) PROOF . First note that if is a complex number with || < 1 then log e $ ( 1—5 ) = 5—15 + 1 / 252 + 31 53+ . ) .. = 0 ( 52 ) , as 5 → 0. Let ƒ ( 5 ) = 5 ...
Page 1281
... complex - valued function integrable over every compact subinterval of I. Let to I , and let co C1 , C - 1 an arbitrary set of n complex numbers . Then there exists a unique fЄ A " ( I ) such that • be ( a ) rf = g , i d ( b ) f ( to ) ...
... complex - valued function integrable over every compact subinterval of I. Let to I , and let co C1 , C - 1 an arbitrary set of n complex numbers . Then there exists a unique fЄ A " ( I ) such that • be ( a ) rf = g , i d ( b ) f ( to ) ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear 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 transform unique unitary vanishes vector zero