Research Interests

• Analysis: Partial Differential and Integral Equations, Spectral Theory, Functional Analysis, Complex Variables, Microlocal Analysis, Global Analysis

• Applications: Fluid Dynamics, Elasticity Theory

Recent Publications

• Some remarks on the boundedness of pseudodifferential operators, Math. Nachr. 183, 229-273 (1997).
• On two-dimensional singular integral operators with conformal Carleman shift, J. Operator Theory, 37, 263-279 (1997) (with R. Duduchava and A. Saginashvili).
• An algebra of integral operators with fixed singularities in kernels, Integr. Equat. Oper. Th. 33, 406-425 (1999) (with R. Duduchava and N. Krupnik).
• Geometry of higher order relative spectra and projection methods, J. Operator Theory, 44, 43-62 (2000).
• Semi-elliptic operators generated by vector fields, Dissertationes Mathematicae, 409, 96 pp. (2002).
• A Riemann-Hilbert problem and the Bernoulli boundary condition in the variational theory of Stokes waves, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 20, 1, 37-52 (2003) (with J.F. Toland).
• Riemann-Hilbert theory for problems with vanishing coefficients that arise in nonlinear hydrodynamics, J. Funct. Analysis, 197, 1, 283-300 (2003) (with J.F. Toland).
• Quasi-conformal mappings and periodic spectral problems in dimension two, Journal d'Analyse Mathèmatique, 91, 67-104 (2003) (with A.V. Sobolev).
• Spectral pollution and second order relative spectra for self-adjoint operators, IMA J. Numerical Analysis, 24, 3, 393-416 (2004) (with M. Levitin).
• The inner variation of an operator and the essential norms of pointwise multipliers in function spaces, Houston Journal of Mathematics, 31, 3, 841-855 (2005) (with D.E. Edmunds).
• Uniqueness results for the Riemann-Hilbert problem with a vanishing coefficient, Integr. Equat. Oper. Th. 56, 115-127 (2006) (with J.A. Virtanen).
• Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II, Journal of Mathematical Analysis and Applications, 325, 745-751 (2007) (with S. Samko and B.Vakulov).
• A remark on the essential spectra of Toeplitz operators with bounded measurable coefficients, Integr. Equat. Oper. Th. 57, 127-132 (2007).
• Spectra of Toeplitz operators and compositions of Muckenhoupt weights with Blaschke products, Integr. Equat. Oper. Th. 61, 63-75 (2008) (with S. Grudsky).
• On the level sets of the resolvent norm of a linear operator, Bulletin LMS, 40, 3, 493-504 (2008).
• Bernoulli free-boundary problems, Memoirs AMS, 196, 914, 70 pp. (2008) (with J.F. Toland).
• On the definition of pseudospectra, Bulletin LMS, 41, 3, 524-534 (2009).
• The level sets of the resolvent norm and convexity properties of Banach spaces, Archiv der Mathematik, 93, 1, 59-66 (2009) (with S. Shkarin).
• Toeplitz matrices with variable coefficients, pseudodifferential operators, and Strichartz's method, Math. Nachr. 283, 1, 126-138 (2010).
• Pseudospectra of semigroup generators, Bulletin LMS, 42, 6, 1031-1034 (2010).
• On some open problems in spectral theory, Operator Theory and Its Applications. In Memory of V. B. Lidskii (1924-2008). Amer. Math. Soc. Transl. 231, 173-180 (2010).
• Applications of Blaschke products to the spectral theory of Toeplitz operators, In: J. Mashreghi and E. Fricain, Blaschke Products and Their Applications. Fields Institute Communications, 1--30 (2012) (with S. Grudsky).
• On the limit behaviour of second order relative spectra of self-adjoint operators, Journal of Spectral Theory, (to appear).
• On the Amick-Fraenkel conjecture, Quarterly J. Math. (to appear).
• An estimate for the Morse index of a Stokes wave, Arch. Rational Mech. Anal. (to appear).