Linear Operators: Spectral theory |
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Page 1305
... considering some simple examples of differential operators . The simplest example of a formally self adjoint differential operator is the operator 7 = i ( d / dt ) . We shall consider three choices for the interval I. = Case 1 : I [ 0 ...
... considering some simple examples of differential operators . The simplest example of a formally self adjoint differential operator is the operator 7 = i ( d / dt ) . We shall consider three choices for the interval I. = Case 1 : I [ 0 ...
Page 1408
... Consider the B - space CB [ x , b ) of all bounded continuous functions defined on [ x , b ) , with norm [ f ] = lubst < lf ( t ) . In this space consider the following linear transformation : ( Mƒ ) ( t ) = √ , * sin ( s — t ) q ( s ) ...
... Consider the B - space CB [ x , b ) of all bounded continuous functions defined on [ x , b ) , with norm [ f ] = lubst < lf ( t ) . In this space consider the following linear transformation : ( Mƒ ) ( t ) = √ , * sin ( s — t ) q ( s ) ...
Page 1682
... consider the convolution ( iv ) - c ( u ) = √à „ a ( x ) b ( x — u ) du . If b is in L1 ( E " ) then Lemma XI.3.1 shows that equation ( iv ) defines a continuous map a → c from L1 ( E ” ) into L1 ( E " ) as well as one from L ( E ...
... consider the convolution ( iv ) - c ( u ) = √à „ a ( x ) b ( x — u ) du . If b is in L1 ( E " ) then Lemma XI.3.1 shows that equation ( iv ) defines a continuous map a → c from L1 ( E ” ) into L1 ( E " ) as well as one from L ( E ...
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