Linear Operators, Part 2 |
From inside the book
Results 1-3 of 71
Page 1307
... boundary values at a and D1 , D2 are boundary values at b , such that ( Tf , g ) - ( f , Tg ) - C1 ( ƒ ) С2 ( g ) —С2 ( f ) C1 ( g ) + D1 ( ƒ ) D2 ( g ) —D2 ( f ) D1 ( g ) , ƒ , g = D ( T1 ( t ) ) . PROOF . Let A be any boundary value ...
... boundary values at a and D1 , D2 are boundary values at b , such that ( Tf , g ) - ( f , Tg ) - C1 ( ƒ ) С2 ( g ) —С2 ( f ) C1 ( g ) + D1 ( ƒ ) D2 ( g ) —D2 ( f ) D1 ( g ) , ƒ , g = D ( T1 ( t ) ) . PROOF . Let A be any boundary value ...
Page 1310
... boundary conditions at a , and exactly one solution y ( t , λ ) of ( t −2 ) y square - integrable at b and satisfying the boundary conditions at b . - 0 PROOF . We shall show the theorem is true in each of the four cases discussed ...
... boundary conditions at a , and exactly one solution y ( t , λ ) of ( t −2 ) y square - integrable at b and satisfying the boundary conditions at b . - 0 PROOF . We shall show the theorem is true in each of the four cases discussed ...
Page 1471
... boundary values at b , we may find two real boundary values D1 , D2 for T2 at b , such that 2 2 ( T2f , g ) — ( f , T2g ) = D1 ( ƒ ) D2 ( g ) —D2 ( ƒ ) D1 ( g ) —F , ( f , g ) , f , g € D ( T1 ( TM 2 ) ) . By Theorem 2.30 and Corollary ...
... boundary values at b , we may find two real boundary values D1 , D2 for T2 at b , such that 2 2 ( T2f , g ) — ( f , T2g ) = D1 ( ƒ ) D2 ( g ) —D2 ( ƒ ) D1 ( g ) —F , ( f , g ) , f , g € D ( T1 ( TM 2 ) ) . By Theorem 2.30 and Corollary ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Copyright | |
57 other sections not shown
Other editions - View all
Common terms and phrases
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