Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "
On a weighted singular integral operator with shifts and slowly oscillating data."
Complex Analysis and Operator Theory. 10.6 (2016): 1101-1131.
AbstractLet \(\alpha,\beta\) be orientation-preserving diffeomorphism (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\) and \(U_\alpha,U_\beta\) be the isometric shift operators on \(L^p(\mathbb{R}_+)\) given by \(U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)\), \(U_\beta f=(\beta')^{1/p}(f\circ\beta)\), and \(P_2^\pm=(I\pm S_2)/2\) where \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \]
is the weighted Cauchy singular integral operator. We prove that if \(\alpha',\beta'\) and \(c,d\) are continuous on \(\mathbb{R}_+\) and slowly oscillating at \(0\) and \(\ infty\), and \[ \limsup_{t\to s}|c(t)|<1,\quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] then the operator \((I-cU_\alpha)P_2^++(I-dU_\beta)P_2^-\) is Fredholm on \(L^p(\mathbb{R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.
Karlovich, Alexei Yu. "
Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras."
Modern Analysis and Applications: Mark Krein Centenary Conference, Vol. 1. Operator Theory Advances and Applications, 190. Eds. V. Adamyan, Y. Berezansky, I. Gohberg, M. Gorbachuk, V. Gorbachuk, A. Kochubei, H. Langer, and G. Popov. Basel: Birkhäuser, 2009. 341-359.
AbstractLet \(\alpha,\beta\in(0,1)\) and
\[
K^{\alpha,\beta}:=\left\{a\in L^\infty(\mathbb{T}):\
\sum_{k=1}^\infty |\widehat{a}(-k)|^2 k^{2\alpha}<\infty,\
\sum_{k=1}^\infty |\widehat{a}(k)|^2 k^{2\beta}<\infty
\right\}.
\]
Mark Krein proved in 1966 that \(K^{1/2,1/2}\) forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended
earlier results of Gabor Szegö for scalar symbols and established the asymptotic trace formula
\[
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)
\quad\text{as}\ n\to\infty
\]
for finite Toeplitz matrices \(T_n(a)\) with matrix symbols \(a\in K^{1/2,1/2}_{N\times N}\). We show that if \(\alpha+\beta\ge 1\) and \(a\in K^{\alpha,\beta}_{N\times N}\), then the Szegö-Widom asymptotic trace formula holds with \(o(1)\) replaced by \(o(n^{1-\alpha-\beta})\).
Karlovich, Alexei Yu. "
Singular integral operators with flip and unbounded coefficients on rearrangement-invariant spaces."
Functional Analysis and its Applications. Proceedings of the international conference, dedicated to the 110th anniversary of Stefan Banach, Lviv National University, Lviv, Ukraine, May 28--31, 2002. Eds. V. Kadets, and W. Zelazko. Amsterdam: Elsevier, 2004. 123-131.
AbstractWe prove Fredholm criteria for singular integral operators of the form
\[
P_++M_bP_-+M_uUP_-,
\]
where \(P_\pm\) are the Riesz projections, \(U\) is the flip operator, and \(M_b,M_u\) are operators of multiplication by functions \(b,u\), respectively, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that \(M_b\) is bounded, but \(M_u\) may be unbounded. The function \(u\) belongs to a class of, in general, unbounded functions that relates to the Douglas algebra \(H^\infty+C\).
Karlovich, Alexei Yu., and Yuri I. Karlovich. "
Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts."
Factorization, Singular Operators and Related Problems. Eds. Stefan Samko, Amarino Lebre, and António Ferreira dos Santos. Dordrecht: Kluwer Academic Publishers, 2003. 111-129.
AbstractThe paper is devoted to the compactness of the commutators \(aS_\Gamma - S_\Gamma aI\) and \(W_\alpha S_\Gamma - S_\Gamma W_\alpha\), where \(S_\Gamma\) is the Cauchy singular integral operator, \(a\) is a bounded measurable function, \(W_\alpha\) is the shift operator given by \(W_\alpha f = f\circ\alpha\), and \(\alpha\) is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively, the derivative of the shift a has vanishing mean oscillation.
Karlovich, Alexei Yu., and Yuri I. Karlovich. "
Invertibility in Banach algebras of functional operators with non-Carleman shifts."
Ukrains'kyj matematychnyj kongres -- 2001. Pratsi. Sektsiya 11. Funktsional'nyj analiz. Kyiv: Instytut Matematyky NAN Ukrainy, 2002. 107-124.
AbstractWe prove the inverse closedness of the Banach algebra \(\mathfrak{A}_p\) of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on \(L^p\). We suppose that \(1 \le p \le \infty\) and the generators of the algebra \(\mathfrak{A}_p\) have essentially bounded data. An invertibility criterion for functional operators in \(\mathfrak{A}_p\) is obtained in terms of
the invertibility of a family of discrete operators on \(l^p\). An effective invertibility criterion is established for binomial difference operators with \(l^\infty\) coefficients on the spaces \(l^p\). Using the reduction to binomial difference operators, we give effective criteria of invertibility for binomial functional operators on the spaces \(L^p\).
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "
Invertibility of functional operators with slowly oscillating non-Carleman shifts."
Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, 142. Eds. Albrecht Böttcher, Marinus A. Kaashoek, Amarino Brites Lebre, António Ferreira dos Santos, and Frank-Olme Speck. Basel: Birkhäuser, 2003. 147-174.
AbstractWe prove criteria for the invertibility of the binomial functional operator
\[
A=aI-bW_\alpha
\]
in the Lebesgue spaces \(L^p(0,1)\), \( 1 < p < \infty\), where \(a\) and \(b\) are continuous functions on \((0,1)\), \(I\) is the identity operator, \(W_\alpha\) is the shift operator, \(W_\alpha f=f\circ\alpha\), generated by a non-Carleman shift \(\alpha:[0,1]\to[0,1]\) which has only two fixed points \(0\) and \(1\). We suppose that \(\log\alpha'\) is bounded and continuous on \((0,1)\) and that \(a,b,\alpha'\) slowly oscillate at \(0\) and \(1\). The main difficulty connected with slow oscillation is overcome by using the method of limit operators.