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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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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