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, 2021.
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. 46.1 (2021): 121-148.Website
D
Diening, Lars, Oleksiy Karlovych, and Eugene Shargorodsky. "Addendum to "On interpolation of reflexive variable Lebesgue spaces on which the Hardy–Littlewood maximal operator is bounded"." Georgian Mathematical Journal . 30.2 (2023): 211-212.Website
Diening, Lars, Oleksiy Karlovych, and Eugene Shargorodsky. "On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded." Georgian Mathematical Journal . 29.3 (2022): 347-352.Website
F
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Fourier convolution operators with symbols equivalent to zero at infinity on Banach function spaces." Current Trends in Analysis, its Applications and Computation. Eds. P. Cerejeiras, M. Reissig, I. Sabadini, and J. Toft. Springer, 2022. 335-343.
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. "Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers." Banach Journal of Mathematical Analysis. 15 (2021): 29.Website
Fernandes, Cláudio, Oleksiy Karlovych, and Márcio Valente. "On the density of Laguerre functions in some Banach function spaces." Journal of Inequalities and Special Functions. 13.2 (2022): 37-45.Website
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Calkin images of Fourier convolution operators with slowly oscillating symbols." Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Basel: Birkhäuser Basel, 2021. 193-218.
Fernandes, Cláudio, Oleksiy Karlovych, and Samuel Medalha. "Invertibility of Fourier convolution operators with PC symbols on variable Lebesgue spaces with Khvedelidze weights." Journal of Mathematical Sciences. 266.3 (2022): 419-434.Website
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. 175.1 (2021): 49-61.Website
K
Karlovich, Alexei Yu. "Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm." Operators and Matrices. 1.3 (2007): 427-444. AbstractWebsite

Suppose \(\Gamma\) is a Carleson Jordan curve with logarithmic whirl points, \(\varrho\) is a Khvedelidze weight, \(p:\Gamma\to(1,\infty)\) is a continuous function satisfying \(|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|\) for \(|\tau-t|\le 1/2\), and \(L^{p(\cdot)}(\Gamma,\varrho)\) is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with \(N\times N\) matrix piecewise continuous coefficients are Fredholm on \(L_N^{p(\cdot)}(\Gamma,\varrho)\).

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. Abstract

We 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., Yuri I. Karlovich, and Amarino B. Lebre. "The index of weighted singular integral operators with shifts and slowly oscillating data." Journal of Mathematical Analysis and Applications. 450 (2017): 606-630.Website
Karlovich, Alexei. "Toeplitz operators between distinct abstract Hardy spaces." Extended Abstracts Fall 2019. Trends in Mathematics, vol 12. Eds. Abakumov E., Baranov A., Borichev A., Fedorovskiy K., and Ortega-Cerdà J. Cham: Birkhäuser, 2021. 105-112.
Karlovich, Alexei Yu. "Remark on the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights." Journal of Function Spaces and Applications. 7 (2009): 301-311. AbstractWebsite

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.

Karlovich, Alexei Yu. "Norms of Toeplitz and Hankel operators on Hardy type subspaces of rearrangement-invariant spaces." Integral Equations and Operator Theory. 49 (2004): 43-64. AbstractWebsite

We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices.

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. Abstract13_2002_ukrainian_math_congress-kyiv-01.pdf

We 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. "The generalized Cauchy index of some semi-almost periodic functions." Boletín de la Sociedad Matemática Mexicana. 22.2 (2016): 473-485. AbstractWebsite

We compute the generalized Cauchy index of some semi-almost periodic functions, which are important
in the study of the Fredholm index of singular integral operators with shifts and slowly oscillating data.

Karlovich, Alexei, and Eugene Shargorodsky. "Algebras of convolution type operators with continuous data do not always contain all rank one operators." Integral Equations and Operator Theory. 93.2 (2021): 16.Website