*Schedule:*

Dates: October 11, 18, 19, 25 and 26th, 2001

Times: 10:00 am-12:00 pm

Location: The Fields Institute, Room 210

*Short description:*

The course intends to present a new tool for perturbation analysis
of matrices. If a large matrix is perturbed with a low rank matrix the
spectrum can change dramatically and therefore the resolvent, the eigenvalues
being its poles, changes

dramatically if considered as an analytic function. In contrast, if
considered as a meromorphic function the perturbation

is small.

The "tool" *T*_{1} is presented in:

Olavi Nevanlinna, *Growth of operator valued meromorphic
functions*, Annales Academiae Sci.

Fenn. Math., Vol 25, 2000, 3-30.

The original article discusses operators in Hilbert spaces
but the lectures shall concentrate on matrices. It shall be demonstrated
how the tool can be used e.g. in deriving error bounds for Krylov solvers
which are robust in low rank perturbations.

*Preliminary content of lectures*:

- Resolvent of a matrix as a meromorphic function. An example of
a compact operator

with spectrum at origin which transforms to a self adjoint operator
with a rank-

one perturbation.
- (2-3) Two lectures on basics of value distribution theory for scalar
meromorphic

functions. In particular, main properties of the characteristic function
*T*.
- (2-3) Two lectures on basics of value distribution theory for scalar
meromorphic

functions. In particular, main properties of the characteristic function
*T*.
- $T_\infty$ and
*T*_{1}: Two generalizations of *T*
to matrix and operator valued

functions.
- The total logarithmic size $s(A):=\sum \log^+ \sigma_j$ (where
$\sigma_j$ denote

the singular values) of a matrix. Basic properties like behavior when
forming sum,

product, Kronecker product etc. Behavior in similarity transformations.
- Subharmonicity of the total logarithmic size
*s*(A(*z*))
of a matrix valued analytic

function A(*z*). Behavior near possible poles.
- Perturbation results for the resolvent when "small" means small
rank (but not small

norm). Defective eigenvalues (in the sense of linear algebra) are
linked to Picard

exceptional values and to defects in value distribution theory.
- Application to Krylov subspace solvers: convergence bounds which
are robust in low

rank changes of the iteration matrix.
- Applications to power bounded operators, e.g. to the behavior of
*A*^{n+1}-A^{n} to

Kreiss matrix theorem etc.
- What next?

For more details on the thematic year see the link on the

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