Invariant Subspaces of Linear Transformations in Hilbert Space a Survey of 1961
Invariant Subspaces of Linear Transformations in Hilbert Space a Survey of 1961
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V/henever such a familjr of projec- tions is chosen, then for every regular a >, a TP(a) = r P(t)(T-T*)dP(t) with convergence of Stieltjes sums in the trace norm. As a result of this integral representation, we get the estimate t(T)2 < i t(T-T*)2 provided that T is a Vol terra transformation and that T-T* is of trace class. Tine theorems are due to Sahnovic in 1959- The stated formulation is that of Gohberg and Krein in the same year. These results may be summarized broadly by saying that a Volterra transfomiation T is determined by a knowledge of T-T* and of a totally ordered family of invariant subspaces. Actually a completely satisfactory theorem exists only when T-T* is of trace class, but in I961 Macaev announced more general results which relax the hypotheses on T-T* . This aspect of Volterra transfon^iatlons may therefore be considered vjell in control even though a small area for more work is left. For Volterra transformations the major unsolved problem is to determine all the invariant subspaces from a knowledge of anj'- single one-parameter family of invariant subspaces.
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You can find similar books in the "Read Also" column, or choose other free books by Louis De Branges to read online
To quickly assess the difficulty of the text, read a short excerpt:
V/henever such a familjr of projec- tions is chosen, then for every regular a >, a TP(a) = r P(t)(T-T*)dP(t) with convergence of Stieltjes sums in the trace norm. As a result of this integral representation, we get the estimate t(T)2 < i t(T-T*)2 provided that T is a Vol terra transformation and that T-T* is of trace class. Tine theorems are due to Sahnovic in 1959- The stated formulation is that of Gohberg and Krein in the same year. These results may be summarized broadly by saying that a Volterra transfomiation T is determined by a knowledge of T-T* and of a totally ordered family of invariant subspaces. Actually a completely satisfactory theorem exists only when T-T* is of trace class, but in I961 Macaev announced more general results which relax the hypotheses on T-T* . This aspect of Volterra transfon^iatlons may therefore be considered vjell in control even though a small area for more work is left. For Volterra transformations the major unsolved problem is to determine all the invariant subspaces from a knowledge of anj'- single one-parameter family of invariant subspaces.
What to read after Invariant Subspaces of Linear Transformations in Hilbert Space a Survey of 1961?
You can find similar books in the "Read Also" column, or choose other free books by Louis De Branges to read online
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