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Achievements, Fields of Interest
My fields of scientific interests mostly contain theory of semigroups of operators, cosine
operator­function theory, differential equations in Banach spaces, numerical analysis in lin­
ear and semilinear differential equations in abstract spaces, approximation of the spectrum
of operators. Let me describe the most interesting to my mind points of my research.
* The stable and regular consistency of closed linear operators on general discretisation
scheme for approximation of spectrum of operators are investigated. It is shown that for
unbounded linear operators regular consistent implies coincidence of dimensions of general­
ized eigenvector spaces of corresponding eigenvalues. So the eigenvalue is stable in Kato's
sense iff in the neighborhood of this eigenvalue one has regular consistency of operators.
* It is established that compact convergence of resolvents implies regular consistent of
closed operators on the whole complex plane.
* It is developed approach of checking regular convergence of operators by using the
Ahues's type conditions, which are easily applied in practice. For instance, it is shown
in principle that convergence k(Bn \Gamma B)B q
n k ! 0 as n ! 1 for some q implies that
– \Gamma Bn ! – \Gamma B regularly.
* Investigated the simplest difference schemes for solving Cauchy problems for the first
and the second order equations in a Banach space E
u 0 (t) = Au(t); u(0) = u 0 ; (1)
u 00 (t) = Au(t); u(0) = u 0 ; u 0 (0) = u 1 ; (2)
For semidiscrete case, i.e. for approximation in space variable
u 0
n (t) = Anun (t); un (0) = u 0 ;
u 00
n (t) = Anun (t); un (0) = u 0
n ; u 0
n (0) = u 1
n ;
it is established Trotter­Kato's version theorem in the general discretisation scheme for C 0 ­
semigroups and for analytic semigroups. For full discretisation scheme it is well known that
in general for the C 0 ­semigroups and the forward scheme we have the following stability
condition
ÜkA 2
n k = O(1):
In the case of analytic semigroups for the forward scheme obtained the following stability
condition
ÜkAnk ! 1=(M + 2);
which can't be improved even in Hilbert spaces for self­adjoint operators. In the case of
almost periodic semigroups and the forward scheme for differential equation of the first
order in time (1) obtained the stability condition
ÜkAnk ! 1:
To consider second order case (2) we introduce the notion of discrete cosine operator­
functions and develop the theory of convergence of discrete cosine operator functions. It is
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proved that condition ÜkAnk ! 4 is necessary and sufficient for the stability of the forward
difference scheme for solving problem (2) with almost periodic solutions.
* Discovered stability condition of the forward scheme for the positive C 0 ­semigroups
(first order equation in time) in the form ÜkAnk ! 1: In formulation of the Trotter­Kato's
version theorem for positive semigroups instead of stability estimate on semigroups we use
the condition of positive off diagonal property for infinitesimal generators of semigroups.
* For the Cauchy problem in the space C
`(\Omega\Gamma with the parabolic equation in the form
u 0
t (x; t) = \Delta x u(x; t) + f(x; t);
full discretisation schemes are considered. It is established that in general Crank­Nicolson
scheme is not maximal regularly stable for the functions from the set of functions
\Xi = B Ü (0; T ; C `
h(\Omega h )) `` C Ü (0; T ; C
h(\Omega h )). For such functions Crank­Nicolson scheme
is only unconditionally stable. The maximal regularity stability for the functions from \Xi is
proved for the implicit schemes and also for the case of Crank­Nicolson scheme, but under
assumption on relation of steps discretisation in time and space.
* Investigated the property of almost periodicity of cosine (COF) and sine operator­
functions. It is proved that COF C(t; A) is almost periodic iff C(t; A) is uniformly bounded,
oe(A) ` R \Gamma and the set of all linear combinations of eigenvectors of operator A is dense in
E. Each generalized solution of Cauchy problem (2) is almost periodic iff C(t; A) is almost
periodic and 0 2 ae(A):
* Investigated the property of compactness of cosine and sine operator functions. In
particular, it is shown that C(t; A) \Gamma I is compact for every t ? 0 iff the generator A is
compact.
* The asymptotic behavior of COF C(t; A) investigated in terms of the behavior of
resolvent and also behavior of semigroup exp(tA) near imaginary axis. In particular, we have
proved that if kexp(zA)k Ÿ M 1 ( jzj
Rez ) 1=2 for any Rez ? 0, then kC(t; A)xk Ÿ M 2 (1+t 2 )kAxk
for any x 2 D(A):
* Investigated the local C­cosine families. the generation theorem for the local C­
cosine families is obtained. Connections with the local n­times integrated semigroups has
been studied.
* It is shown that the spectral condition 0 2 ae(A) [ Coe(A) is necessary and sufficient
for fi­Cesaro stability of bounded C 0 ­semigroup on a Banach space E: Analogous results
are valid for other averages of semigroups and cosine operator functions.
* Asymptotical behaviour of semigroups (and others resolution families) are considered.
Let A be the generator of a tempered C 0 ­semigroup acting on a Banach space E. Let ff ? 0
and suppose that there exists an operator B 2 B(E) such that
(i) – ff+1 (– \Gamma A) \Gamma1 ! B in the strong operator topology as – ! 0 + in R;
(ii) there exists C ? 0; N – 0 and ae 0 ? 0 such that
kae 1+ff (aee iOE \Gamma A) \Gamma1 k Ÿ C
cos N OE ; 0 ! ae Ÿ ae 0 ; OE 2 (\Gammaú=2; ú=2):
Conditions (i) and (ii) are necessary and sufficient for existence of positive integer m
such that
lim
t!1
1
t m+ff
Z t
0
(t \Gamma s) m\Gamma1 e As ds = B
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in the strong operator topology.
The same is proved for cosine operator­functions.
* Investigated approximations of the problems (1) and (2) by rational approximation
( they have a more complicated difference schemes, for example, Pade fractions) and the
optimal order of convergence and stability conditions are obtained. In particular, it is shown
that Crank­Nicolson scheme for analytic semigroups is stable in a general Banach space E.
* The Richardson extrapolation methods are investigated for increasing the accuracy
of approximation of the solution of the Cauchy problems (1) and (2) in a Banach space
using diagonal Pade fractions.
* Discretisation of an evolution equation with analytic semigroup and backwards time
(ill­posed problem for (1)) are investigated. We consider regularization methods and obtain
the error estimates of the methods which are can't be improved even in self­adjoint case.
* The approximation of ill­posed Cauchy problem by using C­semigroups is established.
More exactly, if in (1) the operator \GammaA generates analytic semigroups, then the solution is
given by some C­semigroup u(t) = C \Gamma1 S(t)u 0 : We proved Trotter­Kato Theorem for the
local C­semigroups, and some discrete semigroups approximation theorems are established.
It is also shown that stochastic differential equation
du ff (t) = \GammaAu ff (t)dt \Gamma ffAu ff (t)dw(t); u(0) = u 0 ;
where w(\Delta) is a standart Wiener process, gives us stochastic regularisation of (1) with \GammaA
which generates analytic semigroup. Discretisation in time, i.e. simplest time discretisation
scheme, is also considered. More precisely, we established that the scheme
U n;ff (t + Ü n ) \Gamma U n;ff (t) = \GammaÜ nAnU n;ff (t) \Gamma ff \Deltaw(t)A n U n;ff (t); U n;ff (0) = I n ;
is stable and U n;ff n
(t)u 0
n ! u(t) as n !1; i.e.
sup
ku 0
n \Gammap nu 0 kŸffi
kU n;ff n
(t)u 0
n \Gamma pnu(t)k ! 0 as ffi n ! 0:
Stability of the scheme is meant in the sense:
jn := sup \Phi
E \Theta
kU n;ff n
(t)u 0
n \Gamma u n;ff n
(t)k \Lambda
: ku 0
n k Ÿ 1 \Psi
! 0;
and convergence of method is meant in the following sense: E \Theta
kU n;ff n
(t)u 0
n \Gamma pnu(t)k \Lambda
Ÿ
C p
Ü nkA exp(\GammaT A)u 0 k + ku n;ff n
(t) \Gamma pn u ff n
k + Cjnku 0
n k; 0 ! t Ÿ T: Here we suppose
that lim sup n!1 kn ff 2
n ! C 1 t, and there are constant C such that Ü nkA 2
n ke ckA 2
n k Ÿ C:
* Investigated the Cauchy problem
u 0 (t) = Au(t) + f(t; u(t)); u(0) = u 0 ;
with analytic semigroups exp(tA) and function f which is smooth enough in its arguments.
Using theory of rotation of the vector field under condition of compactness of resolvent
we establish the convergence of the difference schemes and obtain order of convergence.
The same is done for the second order equation in time. Under additional assumption
of positiveness in a Banach lattice E or of Ljapunov stability condition one also obtains
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the approximations of the stable periodic solutions of the semilinear equation u 0 (t) =
Au(t) + f(t; u(t)):
* Investigated perturbation of C 0 ­semigroups by multiplicative case. It is shown, in
particular, that the semigroups satisfy kexp(tA) \Gamma exp(tC)k = O(t) as t ! 0 iff C =
A(1 \Gamma – “
U(–)) +– 2 “
U (–) for some Lipschitz continuous step responses function U(t), i.e. the
function which satisfy the equation
U(t + s) \Gamma U(t) = exp(tA)U(s) for t; s – 0; U(0) = 0:
The same is done for cosine operator functions. We also proved that perturbations
with cumulative outputs give us all known additive perturbations of generator of COF.
Perturbations with the step response for COF have not bee considered before at all.
* We investigate also the continuity, measurability, compactness, positivity and almost
periodicity properties of step responses functions and of cumulative outputs functions for
semigroups and COF and also its asymptotic behavior.
* We characterize in terms of semivariation such cosine operator functions C(\Delta) that for
each continuous function f the function R t
0
R t\Gammas
0 C(Ü )f(s)dÜ ds, t ? 0, is twice continuously
differentiable (MR­property). We also give a new characterization of those C(\Delta) which have
bounded generators. It is also proved that if A generates a cosine operator function of
bounded semivariation, then operator A is bounded.
* In Hilbert space H it was established that perturbations in the case of C 0 ­groups and
cosine operator functions are described by step response functions, which have semivaria­
tions CV (U(\Delta); t); CV (F (\Delta); t) of order not less than O(
p
t) as t ! 0:
* It is proved that Cauchy problem for the second order equations in Banach spaces is
coersive in C([0; T ]; E) or in L p ([0; T ]; E) spaces iff operator A is bounded.
* Discrete maximal regularity estimates for parabolic equations with elliptic operator of
second order in C ff
(\Omega\Gamma spaces established. Let us note that in C
ff(\Omega\Gamma space elliptic operator
does not generate C 0 \Gammasemigroup. Consider the following discrete scheme
U n \Gamma U n\Gamma1
Ü = A( U n + U n\Gamma1
2 ) + F n ; n 2 f1; :::; Kg;U 0 = u 0 ;
where operator A generates analytic semigroup, but not C 0 \Gammasemigroup in C
ff(\Omega\Gamma space.
Let G 2 B(f0; 1; :::; N 1 g; ~
E) be such that G 0 = 0 and consider the problem with restriction
F n = RG n for n = 1; :::; N 1 and u 0 = 0. Then there exists a unique solution U 2
B
i
f0; 1; :::; N 1 g; E
j
and for any n = 0; 1; :::; N 1
kU n k2+2` Ÿ c
i
max
0ŸnŸN1
kG n k 2` + max
0Ÿn1!n2ŸN1
((n 2 \Gamma n 1 )Ü ) \Gamma` p(G n2 \Gamma G n1 )
j
;
where p is some seminorm and the norms are discrete analogy of C
ff(\Omega\Gamma space norms.
* Discrete maximal regularity inequality for Cauchy problem in L p
Ün (R; E) space is
considered. For the general Banach space E the inequality with logarithm is obtained.
We established discrete maximal regularity inequality for implicite scheme and also for
Crank­Nicolson scheme in case of E to be UMD space under condition of R­boundedness
of resolvent.
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* Developed approach to approximate the perturbed C 0 ­semigroups and cosine operator
functions (analogy of Trotter­Kato theorem) using infinitesimal properties of step responses
and cumulative outputs functions.
* Developed approach to approximate the perturbed C 0 ­semigroups and cosine operator
functions by discretisation in time using information on step responses and cumulative
outputs functions near the zero or using the information on Laplace transform of them.
* Investigated the method of conjugate gradients for adaptive spatial filtering. First
we find the proper preprocessor, which reduces the dimensions of the problem in two times
without increasing the conditionally number. The second, a comparison of the proposed
adaptive processing technique with the existing gradient methods has shown that the rate
of convergence exceeds the rate of convergence of ordinary gradient algorithms by a factor
of more than 2.
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