Linear Operators, Part 2 |
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Page 1099
... consider the case in which the Hilbert space is finite - dimensional . The argument in this special case is as follows . Since both sides . of ( 1 ) are continuous in T and since every finite matrix may be ap- proximated arbitrarily ...
... consider the case in which the Hilbert space is finite - dimensional . The argument in this special case is as follows . Since both sides . of ( 1 ) are continuous in T and since every finite matrix may be ap- proximated arbitrarily ...
Page 1155
... consider the special form that the results of Sections 3 and 4 take when R is a compact Abelian group , a case studied ex- plicitly in Section 1 . 8 THEOREM . If R is a compact Abelian group , its character group R is discrete ...
... consider the special form that the results of Sections 3 and 4 take when R is a compact Abelian group , a case studied ex- plicitly in Section 1 . 8 THEOREM . If R is a compact Abelian group , its character group R is discrete ...
Page 1305
... considering some simple examples of differential operators . The simplest example of a formally self adjoint differential operator is the operator Ti ( d / dt ) . We shall consider three choices for the interval I. - Case 1 : I [ 0 , 1 ] ...
... considering some simple examples of differential operators . The simplest example of a formally self adjoint differential operator is the operator Ti ( d / dt ) . We shall consider three choices for the interval I. - Case 1 : I [ 0 , 1 ] ...
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