Yuri Tomilov

Fine scales of decay of operator semigroups

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus, and complex, real and harmonic analysis. It also leads to several results of independent interest.

On the Witten index in terms of spectral shift functions

AbstractWe study the model operator DA = (d/dt) + A in L2(R;H) associated with the operator path {A(t)}t=−∞∞, where (Af)(t) = A(t)f(t) for a.e. t ∈ R, and appropriate f ∈ L2(R;H) (with H a separable, complex Hilbert space). Denoting by A± the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A-, and no discrete spectrum assumptions are made on A±. Introducing H1 = DA*DA, H2 = DADA*, the resolvent and semigroup regularized Witten indices of DA, denoted by Wr(DA) and Ws(DA), are defined by

Stability of C~0-semigroups and geometry of Banach spaces

\begin{gathered} {W_r}\left( {{D_A}} \right) = \mathop {lim}\limits_{\lambda \uparrow 0} \left( { - \lambda } \right)t{r_{{L^2}\left( {\mathbb{R};H} \right)}}\left( {{{\left( {{H_1} - \lambda I} \right)}^{ - 1}} - {{\left( {{H_2} - \lambda I} \right)}^{ - 1}}} \right), \hfill \\ {W_s}\left( {{D_A}} \right) = \mathop {lim}\limits_{t \uparrow \infty } t{r_{{L^2}\left( {\mathbb{R};H} \right)}}\left( {{e^{ - t{H_1}}} - {e^{ - t{H_2}}}} \right), \hfill \\ \end{gathered}

SOME OPERATOR BOUNDS EMPLOYING COMPLEX INTERPOLATION REVISITED

Wr(DA)=limλ↑0(−λ)trL2(ℝ;H)((H1−λI)−1−(H2−λI)−1),Ws(DA)=limt↑∞trL2(ℝ;H)(e−tH1−e−tH2), whenever these limits exist. These regularized indices coincide with the Fredholm index of DA whenever the latter is Fredholm.In situations where DA ceases to be a Fredholm operator in L2(R;H) we compute its resolvent (resp., semigroup) regularizedWitten index in terms of the spectral shift function ξ(•; A+, A-) associated with the pair (A+, A-) as follows: Assuming 0 to be a right and a left Lebesgue point of ξ(•; A+, A-), denoted by ξL(0+; A+, A-) and ξL(0-; A+, A-), we prove that 0 is also a right Lebesgue point of ξ(•; H2, H1), denoted by ξL(0+; H2, H1), and that

Analytic continuation and stability of operator semigroups

\begin{array}{*{20}{c}} {{W_r}\left( {{D_A}} \right)}&\begin{gathered} = {W_s}\left( {{D_A}} \right) \hfill \\ = {\xi _L}\left( {{0_ + };{H_2},{H_1}} \right) \hfill \\ = {{\left[ {{\xi _L}\left( {{0_ + };{A_ + },{A_ - }} \right) + {\xi _L}\left( {{0_ - };{A_ + },{A_ - }} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {{\xi _L}\left( {{0_ + };{A_ + },{A_ - }} \right) + {\xi _L}\left( {{0_ - };{A_ + },{A_ - }} \right)} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}, \hfill \\ \end{gathered} \end{array}

Boundary uniqueness of harmonic functions and spectral subspaces of operator groups

Wr(DA)=Ws(DA)=ξL(0+;H2,H1)=[ξL(0+;A+,A−)+ξL(0−;A+,A−)]/2, the principal result of this paper.In the special case where dim(H) < ∞, we prove that the Witten indices of DA are either integer, or half-integer-valued.