COUNTEREXAMPLES IN OPERATOR THEORY |
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PROBLEMS | SOLUTIONS |
OT1. An operator of index zero which isn't invertible. |
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OT2. A compact operator with no eigenvalues. |
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OT3. A week-operator closed subalgebra B of bounded operators on a Hilbert space H such that B B", where B" denotes the doubel commutant of B. |
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OT4. A unitary operator U acting on a Hilbert space whose spectrum is C = {z e C; |z| = 1 }. |
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OT5. An unbounded symmetric operator on an inner product space. |
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OT6. Two selfadjoint operators T and S on a Hilbert space such that sp(ST) is not a subset of R. |
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OT7. Two Hermetian operators T and S on a Hilbert space such that S 0 and -S T S but not |T| S. |
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OT8. A selfadjoint operator T 0 on a Hilbert space such that T is neither positive nor negative. |
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OT9. A bounded operator on a Hilbert space which has no square root. |
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OT10. A bounded increasing sequence of self-adjoint operators on a Hilbert space which is not uniformly convergent. |
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OT11. Given a compact subset K of C, there exists a bounded operator T on a Hilbert space such that sp(T) = K and the set of eigenvalues of T is dense in K. |
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OT12. Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H. |
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OT13. Given a compact subset K of C being the closure of its interior, there exists an operator T acting on a Hilbert space H such that sp(T) = K and T has no eigenvalue. |
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OT14. An operator T on a Hilbert space such that the set eig(T) of all eigenvalues of T is empty but sp(T) f. |
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OT15. A sequence of quasi-nilpotent operators acting on a Hilbert space with a norm limit whose spectral radius is 1. |
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OT16. A sequence of nilpotent operators on H which converges with respect to the norm topology on B(H) to an operator which is not topologically nilpotent. |
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OT17.
(a) A Banach space X and an operator T e B(X) having no nontrivial invariant subspace. (b) A Banach space X and an operator T e B(X) having a nontrivial invariant subspace. |
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OT18.
(a) An injective operator on a Hilbert space H such that the range of T, R(T), isn't dense in H. (b) An operator S that is surjective but Ker(S) {0}. |
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OT19. Two positive operators T S acting on a Hilbert sace such that S^{2} does not majorize T^{2}. |
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OT20. An unbounded operator on a Hilbert space H annihilating an orthonormal basis of H. |
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OT21. An operator U on a Hilbert space, other than I, such that sp(U) = {1} and ||U|| = 1. |
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OT22.
A Hilbert space H such that on B(H) (i) the involution isn't continuous with respect to the strong operator topology; (ii) the weak operator topology and the strong operator topology are different; (iii) the operator norm is not continuous with respect to the strong operator topology and so the weak operator topology; (iv) the weak operator topology and the strong operator topology aren't metrizable; (v) the operation multiplication is continuous in neither weak nor strong operator topology. |
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