| COUNTEREXAMPLES IN OPERATOR THEORY |
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| COUNTEREXAMPLES IN OPERATOR THEORY |
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| PROBLEMS | SOLUTIONS |
| OT1. An operator of index zero which isn't invertible. |
 OT1.dvi OT1.htm OT1.ps OT1.pdf |
| OT2. A compact operator with no eigenvalues. |
OT2.dviOT2.htmOT2.psOT2.pdf |
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.
|
OT3.dviOT3.htmOT3.psOT3.pdf |
| OT4. A unitary operator U acting on a Hilbert space whose spectrum is C = {z e C; |z| = 1 }. |
OT4.dviOT4.htmOT4.psOT4.pdf |
| OT5. An unbounded symmetric operator on an inner product space. |
OT5.dviOT5.htmOT5.psOT5.pdf |
| OT6. Two selfadjoint operators T and S on a Hilbert space such that sp(ST) is not a subset of R. |
OT6.dviOT6.htmOT6.psOT6.pdf |
OT7.
Two Hermetian operators T and S on a Hilbert space such that S  0 and -S  T S but not |T| S.
|
OT7.dviOT7.htmOT7.psOT7.pdf |
| OT8.
A selfadjoint operator T 0 on a Hilbert space such that T is neither positive nor negative.
|
OT8.dviOT8.htmOT8.psOT8.pdf |
| OT9. A bounded operator on a Hilbert space which has no square root. |
OT9.dviOT9.htmOT9.psOT9.pdf |
| OT10. A bounded increasing sequence of self-adjoint operators on a Hilbert space which is not uniformly convergent. |
OT10.dviOT10.htmOT10.psOT10.pdf |
| 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. |
OT11.dviOT11.htmOT11.psOT11.pdf |
| OT12. Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H. |
OT12.dviOT12.htmOT12.psOT12.pdf |
| 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. |
OT13.dviOT13.htmOT13.psOT13.pdf |
| 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.
|
OT14.dviOT14.htmOT14.psOT14.pdf |
| OT15. A sequence of quasi-nilpotent operators acting on a Hilbert space with a norm limit whose spectral radius is 1. |
OT15.dviOT15.htmOT15.psOT15.pdf |
| 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. |
OT16.dviOT16.htmOT16.psOT16.pdf |
| 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. |
OT17.dviOT17.htmOT17.psOT17.pdf |
| 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}.
|
OT18.dviOT81.htmOT18.psOT18.pdf |
| OT19.
Two positive operators T S acting on a Hilbert sace such that S2 does not majorize
T2.
|
OT19.dviOT19.htmOT19.psOT19.pdf |
| OT20. An unbounded operator on a Hilbert space H annihilating an orthonormal basis of H. |
OT20.dviOT20.htmOT20.psOT20.pdf |
| OT21. An operator U on a Hilbert space, other than I, such that sp(U) = {1} and ||U|| = 1. |
OT21.dviOT21.htmOT21.psOT21.pdf |
| 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. |
OT22.dviOT22.htmOT22.psOT22.pdf |
