Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 909
... meaning of this phrase is given in the following definition . Here and elsewhere the symbol will be used for the direct sum of the Hilbert spaces a ( cf. IV.4.18 and IV.4.19 ) . The ath component of an element x in Za will be denoted by ...
... meaning of this phrase is given in the following definition . Here and elsewhere the symbol will be used for the direct sum of the Hilbert spaces a ( cf. IV.4.18 and IV.4.19 ) . The ath component of an element x in Za will be denoted by ...
Page 1280
... differential operator of order n on the interval I , and fe A " ( I ) , then the expression τf has a definite meaning in a very natural sense : we put ( Tf ) ( t ) = = an ( 1280 XIII.1.1 XIII . ORDINARY DIFFERENTIAL OPERATORS.
... differential operator of order n on the interval I , and fe A " ( I ) , then the expression τf has a definite meaning in a very natural sense : we put ( Tf ) ( t ) = = an ( 1280 XIII.1.1 XIII . ORDINARY DIFFERENTIAL OPERATORS.
Page 1648
... meaning to F for each F in D ( I ) . If , on the other hand , the operator 7 is of order zero , i.e. , is the operation of multiplying by a function a in C °° ( I ) , then Definition 5 assigns a meaning to the product aF for each F in D ...
... meaning to F for each F in D ( I ) . If , on the other hand , the operator 7 is of order zero , i.e. , is the operation of multiplying by a function a in C °° ( I ) , then Definition 5 assigns a meaning to the product aF for each F in D ...
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 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