Publications

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A
André, Carlos, Maria Amélia Bastos, Alexei Yu. Karlovich, Bernd Silbermann, and Ion Zaballa(Eds.) Operator Theory, Operator Algebras, and Matrix Theory. Basel: Birkhäuser, 2018.Website
B
Bastos, Maria Amélia, Luís Castro, and Alexei Yu. Karlovich(eds.) Operator Theory, Functional Analysis and Applications. Basel: Birkhäuser, 2020.
Bini, Dario, Torsten Ehrhardt, Alexei Yu. Karlovich, and Ilya M. Spitkovsky(eds.) Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics. The Albrecht Böttcher Anniversary Volume. Basel: Birkhäuser Basel, 2017.Website
Böttcher, Albrecht, Alexei Yu. Karlovich, and Bernd Silbermann. "Generalized Krein algebras and asymptotics of Toeplitz determinants." Methods of Functional Analysis and Topology. 13.2 (2007): 236-261. AbstractWebsite

We give a survey on generalized Krein algebras \(K_{p,q}^{\alpha,\beta}\) and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that \(K_{2,2}^{1/2,1/2}\) is a Banach algebra. Subsequently, Widom proved the strong Szeg\H{o} limit theorem for block Toeplitz determinants with symbols in \((K_{2,2}^{1/2,1/2})_{N\times N}\) and later two of the authors studied symbols in the generalized Krein algebras \((K_{p,q}^{\alpha,\beta})_{N\times N}\), where \(\lambda:=1/p+1/q=\alpha+\beta\) and \(\lambda=1\). We here extend these results to \(0<\lambda<1\). The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.

C
Canarias, Tiago, Alexei Karlovich, and Eugene Shargorodsky. "Multiplication Is an open bilinear mapping in the Banach algebra of functions of bounded Wiener $p$-variation." Real Analysis Exchange (In Press).
F
Fernandes, Cláudio A., and Alexei Yu. Karlovich. "Semi-almost periodic Fourier multipliers on rearrangement-invariant spaces with suitable Muckenhoupt weights." Boletín de la Sociedad Matemática Mexicana. 26.3 (2020): 1135-1162.Website
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Noncompactness of Fourier convolution operators on Banach function spaces." Annals of Functional Analysis. 10.4 (2019): 553-561.
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Márcio Valente. "Invertibility of Fourier convolution operators with piecewise continuous symbols on Banach function spaces." Transactions of A. Razmadze Mathematical Institute (In Press).
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Fourier convolution operators with symbols equivalent to zero at infinity on Banach function spaces." Proceedings of ISAAC 2019. In Press.
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri. I. Karlovich. "Algebra of convolution type operators with continuous data on Banach function spaces." Banach Center Publications. 119 (2019): 157-171.Website
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Calkin images of Fourier convolution operators with slowly oscillating symbols." Proceedings of IWOTA 2019. In Press.
K
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness." Communications in Mathematical Analysis. 17.2 (2014): 189-208. AbstractWebsite

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) of slowly oscillating functions of limited smoothness introduced in [K09]. We show that if \(\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) does not degenerate on the ``boundary" of \(\mathbb{R}_+\times\mathbb{R}\) in a certain sense, then the Mellin PDO \(\operatorname{Op}(\mathfrak{a})\) is Fredholm on the space \(L^p\) for \(p\in(1,\infty)\) and each its regularizer is of the form \(\operatorname{Op}(\mathfrak{b})+K\) where \(K\) is a compact operator on \(L^p\) and \(\mathfrak{b}\) is a certain explicitly constructed function in the same algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) such that \(\mathfrak{b}=1/\mathfrak{a}\) on the ``boundary" of \(\mathbb{R}_+\times\mathbb{R}\). This result complements the known Fredholm criterion from [K09] for Mellin PDO's with symbols in the closure of \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) and extends the corresponding result by V.S. Rabinovich (see [R98]) on Mellin PDO's with slowly oscillating symbols in \(C^\infty(\mathbb{R}_+\times\mathbb{R})\).

Karlovich, Alexei Yu. "Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves." Topics in Operator Theory: Operators, Matrices and Analytic Functions, Vol. 1. Operator Theory: Advances and Applications, 202. Eds. JA Ball, V. Bolotnikov, JW Helton, L. Rodman, and IM Spitkovsky. Basel: Birkhäuser, 2010. 321-336. Abstract

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces \(L^p(\Gamma)\) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces \(L^{p(\cdot)}(\Gamma)\) where \(p:\Gamma\to(1,\infty)\) satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.

Karlovich, Alexei Yu., and Eugene Shargorodsky. "More on the density of analytic polynomials in abstract Hardy spaces." The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, vol. 268. Eds. Albrecht Böttcher, Daniel Potts, Peter Stollman, and David Wenzel. Basel: Birkhäuser, 2018. 319-329.
Karlovich, Alexei Yu. "Wavelet bases in Banach function spaces." Bulletin of the Malaysian Mathematical Sciences Society (In Press).Website
Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "The Cauchy singular integral operator on weighted variable Lebesgue spaces." Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236. Eds. Manuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, and Sergei Treil. Basel: Birkhäuser, 2014. 275-291. Abstract

Let \(p:\mathbb{R}\to(1,\infty)\) be a globally log-Hölder continuous variable exponent and \(w:\mathbb{R}\to[0,\infty]\) be a weight. We prove that the Cauchy singular integral operator \(S\) is bounded on the weighted variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R},w)=\{f:fw\in L^{p(\cdot)}(\mathbb{R})\}\) if and only if the weight \(w\) satisfies $$ \sup_{-\infty < a < b < \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty \quad (1/p(x)+1/p'(x)=1). $$

Karlovich, Alexei Yu. "Density of analytic polynomials in abstract Hardy spaces." Commentationes Mathematicae. 57.2 (2017): 131-141.Website
Karlovich, Alexei Yu. "Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces." Mathematische Nachrichten. 179 (1996): 187-222. AbstractWebsite

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces \(L_M(\Gamma)\), which are generalizations of the Lebesgue spaces \(L_p(\Gamma)\), \(1 < p < \infty\). We suppose that \(\Gamma\) belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient \(G\), we establish a Fredholm criterion and an index formula in terms of the essential range of \(G\) complemented by spiralic horns depending on the Boyd indices of \(L_M(\Gamma)\) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients on \(L_M^n(\Gamma)\).

Karlovich, Alexei Yu., and Andrei K. Lerner. "Commutators of singular integrals on generalized Lp spaces with variable exponent." Publicacions Matematiques. 49.1 (2005): 111-125. AbstractWebsite

A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized Lp spaces with variable exponent.

Karlovich, Alexei Yu. "Some algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces." Operator Theoretical Methods and Applications to Mathematical Physics. The Erhard Meister Memorial Volume. Operator Theory: Advances and Applications, 147. Eds. Israel Gohberg, Wolfgang Wendland, António Ferreira dos Santos, Frank-Ollme Speck, and Francisco Sepúlveda Teixeira. Basel: Birkhäuser, 2004. 287-296. Abstract

In this paper, the author proves that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to \(\ell^1\cap\ell^\Phi_{\varphi,w}\) (resp. to \(\ell^1\cap\ell^\Psi_{\psi,\varrho}\)), where \(\ell^\Phi_{\varphi,w}\) and \(\ell^\Psi_{\psi,\varrho}\) are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences
\[
\varphi=\{\varphi_n\},\quad
\psi=\{\psi_n\},\quad
w=\{w_n\},\quad
\varrho=\{\varrho_n\}
\]
increase and satisfy the \(\Delta_2\)-condition.

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data." Journal of Integral Equations and Applications. 29.3 (2017): 365-399.
Karlovich, Alexei Yu. "Asymptotics of determinants and traces of Toeplitz matrices with symbols in weighted Wiener algebras." Zeitschrift für Analysis und ihre Anwendungen. 26.1 (2007): 43-56. AbstractWebsite

We prove asymptotic formulas for determinants and traces of finite block Toeplitz matrices with symbols belonging to Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. The remainders in these formulas depend on the weights and go rapidly to zero for very smooth symbols. These formulas refine or extend some previous results by Szegö, Widom, Bottcher, and Silbermann.

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. Abstract

The 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. "Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals." The New York Journal of Mathematics. 8 (2002): 215-234. AbstractWebsite

We construct a presymbol for the Banach algebra \(\operatorname{alg}(\Omega, S)\) generated by the Cauchy singular integral operator \(S\) and the operators of multiplication by functions in a Banach subalgebra \(\Omega\) of \(L^\infty\). This presymbol is a homomorphism \(\operatorname{alg}(\Omega,S)\to\Omega\oplus\Omega\) whose kernel coincides with the commutator ideal of \(\operatorname{alg}(\Omega,S)\). In terms of the presymbol, necessary conditions for Fredholmness of an operator in \(\operatorname{alg}(\Omega,S)\) are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.