Linear operators: Spectral operators |
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Page 2397
The theorem will follow as soon as it is shown that the hypotheses of Theorem
XVIII. 2. 34 are ... Hypothesis (ii) has been established and is given by Lemma 7 (
iv). It therefore only remains to establish hypothesis (iii) of Theorem XVIII. 2. 34.
The theorem will follow as soon as it is shown that the hypotheses of Theorem
XVIII. 2. 34 are ... Hypothesis (ii) has been established and is given by Lemma 7 (
iv). It therefore only remains to establish hypothesis (iii) of Theorem XVIII. 2. 34.
Page 2401
Using hypothesis (c), we may write this last equation as (5) <p{B-ftB,A1))=9(A1).
Now, by hypothesis, the map B-*<p(B, Aħ) of 91 -> 91 has norm at most M2(M1 +
M2)~l. Thus, by Lemma VII. 3.4, the equation (6) B-j(B,A1)=A1 has a solution B e
...
Using hypothesis (c), we may write this last equation as (5) <p{B-ftB,A1))=9(A1).
Now, by hypothesis, the map B-*<p(B, Aħ) of 91 -> 91 has norm at most M2(M1 +
M2)~l. Thus, by Lemma VII. 3.4, the equation (6) B-j(B,A1)=A1 has a solution B e
...
Page 2449
On the other hand, assuming the truth of our assertion for a given n, it follows
from (6) that tn + 1 ^ t^ + 6M2tf. Since, by hypothesis, 6M2t1 ^ 1, we have 'n + 1 ^
2<i and our assertion follows. It also follows from (5) and hypotheses (d) and (e)
that ...
On the other hand, assuming the truth of our assertion for a given n, it follows
from (6) that tn + 1 ^ t^ + 6M2tf. Since, by hypothesis, 6M2t1 ^ 1, we have 'n + 1 ^
2<i and our assertion follows. It also follows from (5) and hypotheses (d) and (e)
that ...
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Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
47 other sections not shown
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adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero
Bibliographic information
| Title | Linear operators: Spectral operators Volume 7 of Pure and applied mathematics Wiley classics library Volume 7 of Pure and applied mathematics : a series of texts and monographs Part 3 of Linear Operators, Jacob T. Schwartz |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Wiley-Interscience, 1971 |
| Original from | the University of Michigan |
| Digitized | 2 Feb 2010 |
| ISBN | 0471226394, 9780471226390 |
| Length | 20 pages |
| Subjects | › › Mathematics / Algebra / Linear |
| Export Citation | BiBTeX EndNote RefMan |