| Tài liệu tham khảo |
Loại |
Chi tiết |
| 6. Let G be an A 2-inner function. With the notation from the previous problem, show that JG either has index 1 or 2 |
Sách, tạp chí |
| Tiêu đề: |
A "2-inner function. With the notation from the previous problem, show that "JG |
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| 7. A maximal inner space in A 2 is an inner space contained in no larger inner space. Show that every inner space is contained in a maximal inner space.Hint: apply Zorn's lemma |
Sách, tạp chí |
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| 8. If G is an A 2 -inner function, then the one-dimensional space generated by G is a maximal inner space if and only if JG has the index 1, where JG is as defined in Problem 5 |
Sách, tạp chí |
| Tiêu đề: |
JG "has the index 1, where "JG |
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| 9. For an invariant subspace I in A 2 , let Mz[I] denote the multiplication oper- ator on I induced by the coordinate function z. Show that Mz[I] and Mz[J]are unitarily equivalent if and only if I = J |
Sách, tạp chí |
| Tiêu đề: |
I "in "A"2 , "let "Mz[I] "denote the multiplication oper-ator on "I "induced by the coordinate function "z. "Show that "Mz[I] "and "Mz[J] "are unitarily equivalent if and only if |
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| 10. Let I = A2 and J be an invariant subspace of A2. Show that Mz[I] and Mz [J] are similar if and only if J is generated by a Blaschke product whose zero set is the union of finitely many interpolating sequences. See [29] |
Sách, tạp chí |
| Tiêu đề: |
I = A2 "and "J "be an invariant subspace of "A2. "Show that "Mz[I] "and "Mz [J] "are similar if and only if "J |
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| 11. Let 1= A2, and let J be an invariant subspace of A2. Show that Mz[I] and Mz[J] are quasi-similar if and only if J is generated by a bounded analytic function. See [70] |
Sách, tạp chí |
| Tiêu đề: |
1= A2, "and let "J "be an invariant subspace of "A2. "Show that "Mz[I] "and "Mz[J] "are quasi-similar if and only if "J |
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| 12. For any positive real number a, let Ia be the invariant subspace of A2 generated by the singular inner function Sa (with a single point mass a atz = 1). Show that Mz[Ia] and Mz[Ir] are similar for all positive a and T.See [141] |
Sách, tạp chí |
| Tiêu đề: |
a, "let "Ia "be the invariant subspace of "A2 "generated by the singular inner function "Sa "(with a single point mass "a "at "z "= 1). Show that "Mz[Ia] "and "Mz[Ir] "are similar for all positive "a "and "T |
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| 13. Show that if the invariant subspace I of A 2 is singly generated or if I is zero-based, then I has the index 1 |
Sách, tạp chí |
| Tiêu đề: |
I "of "A "2 is singly generated or if "I "is zero-based, then "I |
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| 14. If A and B are disjoint regular sequences, then A U B is regular, and the decomposition A U B is homogeneous |
Sách, tạp chí |
| Tiêu đề: |
A "and "B "are disjoint regular sequences, then "A "U "B "is regular, and the decomposition "A "U "B |
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| 15. If I and J are invariant subspaces in A~ of index 1, with the properties that I C J and n = dim(J / I) < +00, then there exists a Blaschke product b with n zeros such that 1= bJ. What if I, J have higher index, say 2 |
Sách, tạp chí |
| Tiêu đề: |
I "and "J "are invariant subspaces in A~ of index 1, with the properties that "I "C "J "and "n "= dim(J / "I) "< +00, then there exists a Blaschke product "b "with "n "zeros such that "1= bJ. "What if "I, J |
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| 16. Let A be a positive bounded operator on the (separable) Hilbert space 1i (over the scalar field C, as usual), which means that {Ax, x} 2: 0 for all x E 1i. Suppose 1i1 is a closed subspace of1i, and that {Ax, x} = 0 for all x E 1i1. Show that Ax = 0 for all x E 1i1 |
Sách, tạp chí |
| Tiêu đề: |
1i "(over the scalar field C, as usual), which means that "{Ax, "x} "2: 0 for all "x "E "1i. "Suppose "1i1 "is a closed subspace "of1i, "and that "{Ax, x} "= 0 for all "x "E "1i1. "Show that "Ax "= 0 for all "x "E |
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| 17. Fix 0 < p < +00 and -1 < a < +00. Recall thatthe index of an invariant subspace I in A~ is defined as the dimension of the quotient space 1/ zI.Show that for)... E JD>, (z - )",)1 is a closed subspace of I, and that the dimension of the quotient space I/(z - )",)1 does not depend on ).... Hint |
Sách, tạp chí |
| Tiêu đề: |
)1 is a closed subspace of I, and that the dimension of the quotient space I/(z - ) |
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