Linear Operators: Spectral theory |
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Page 1279
... C " ( J ) , where J is a compact interval , have been defined in Chapter IV . If ƒ is in C " ( J ) , let us agree to write the norm off as f ( " ) whenever it is desired to emphasize the integer n . The space C °° ( J ) = || C " ( J ) ...
... C " ( J ) , where J is a compact interval , have been defined in Chapter IV . If ƒ is in C " ( J ) , let us agree to write the norm off as f ( " ) whenever it is desired to emphasize the integer n . The space C °° ( J ) = || C " ( J ) ...
Page 1316
... ( c , c - ) - K ( c , c + ) = 0 , - + i = 0 , ... , n − 2 , K ( n - 1 ) ( c , c + ) - K ( n - 1 ) ( c , c− ) = ( −1 ) ” [ a „ ( c ) ] − 1 . Further information on the kernel K is given in the next lemma . 5 LEMMA . The function K ( c ...
... ( c , c - ) - K ( c , c + ) = 0 , - + i = 0 , ... , n − 2 , K ( n - 1 ) ( c , c + ) - K ( n - 1 ) ( c , c− ) = ( −1 ) ” [ a „ ( c ) ] − 1 . Further information on the kernel K is given in the next lemma . 5 LEMMA . The function K ( c ...
Page 1638
... C∞ ( I ) consists of those scalar functions ƒ defined on I which have all partial derivatives of all orders existing and continuous . Similarly , the set C ( I ) consists of those scalar functions defined in I every one of whose ...
... C∞ ( I ) consists of those scalar functions ƒ defined on I which have all partial derivatives of all orders existing and continuous . Similarly , the set C ( I ) consists of those scalar functions defined in I every one of whose ...
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