Linear Operators, Part 2 |
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Page 1186
... R ( T ) and T - ly = x if y Tx . The usual associative laws ( A + B ) + C = A + ( B + C ) and ( AB ) C = A ( BC ) hold , so that sums and products of several operators may be unambiguously written without the use of parentheses . 1186 ...
... R ( T ) and T - ly = x if y Tx . The usual associative laws ( A + B ) + C = A + ( B + C ) and ( AB ) C = A ( BC ) hold , so that sums and products of several operators may be unambiguously written without the use of parentheses . 1186 ...
Page 1440
Nelson Dunford, Jacob T. Schwartz. n b R ( x'f , f ) = Σ [ Rpx ( t ) ] | f ( * ) ( t ) ... R ( tof , f ) ≥ 0 for f = D ( To ( t ) ) . If this is not the case , there ... ly small subinterval [ a 。, b ) of 1440 XIII.7.9 XIII . ORDINARY ...
Nelson Dunford, Jacob T. Schwartz. n b R ( x'f , f ) = Σ [ Rpx ( t ) ] | f ( * ) ( t ) ... R ( tof , f ) ≥ 0 for f = D ( To ( t ) ) . If this is not the case , there ... ly small subinterval [ a 。, b ) of 1440 XIII.7.9 XIII . ORDINARY ...
Page 1441
Nelson Dunford, Jacob T. Schwartz. ly small subinterval [ a 。, b ) of [ a , b ) , we may assume without loss of generality that R ( po ( t ) —c ) ≥ 2. Then R ( Tf , f ) ≥ ( Tcf , f ) +2 ( f , f ) for fe D ( To ( 7 ) ) . It follows ...
Nelson Dunford, Jacob T. Schwartz. ly small subinterval [ a 。, b ) of [ a , b ) , we may assume without loss of generality that R ( po ( t ) —c ) ≥ 2. Then R ( Tf , f ) ≥ ( Tcf , f ) +2 ( f , f ) for fe D ( To ( 7 ) ) . It follows ...
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BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero