Linear Operators: Spectral operators |
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Page 1250
... unique , P is uniquely determined on R ( A ) by the equation of P ( Ax ) Tx . Further the extension of P by continuity from R ( 4 ) to R ( A ) is unique . Since P is zero on R ( A ) 1 it follows that P is uniquely determined by ...
... unique , P is uniquely determined on R ( A ) by the equation of P ( Ax ) Tx . Further the extension of P by continuity from R ( 4 ) to R ( A ) is unique . Since P is zero on R ( A ) 1 it follows that P is uniquely determined by ...
Page 1378
... unique , and matrix measure { ๔กก } , i , j Pij = .... Pij , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . Pii PROOF . Suppose that σ1 , ... , o is a determining set for T. Then it is evident from Theorem 23 that if we define { p1 ...
... unique , and matrix measure { ๔กก } , i , j Pij = .... Pij , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . Pii PROOF . Suppose that σ1 , ... , o is a determining set for T. Then it is evident from Theorem 23 that if we define { p1 ...
Page 1513
... unique solution of the equation with singularities 0 , 1 , ∞ and corresponding exponents 2x , 0 ; 0 , -28 ; 1 / 2 + ya + ฿ , 1 / 2 - y - a + ẞ which is regular and has the value 1 at zero ; the function 22 F is the unique solution of ...
... unique solution of the equation with singularities 0 , 1 , ∞ and corresponding exponents 2x , 0 ; 0 , -28 ; 1 / 2 + ya + ฿ , 1 / 2 - y - a + ẞ which is regular and has the value 1 at zero ; the function 22 F is the unique solution of ...
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 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 linear 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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero