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Some researchers currently active in pseudospectra

Rafikul Alam (perturbation theory)
Constantine Bekas (computation)
Shreemayee Bora (perturbation theory)
Albrecht Böttcher (Toeplitz matrices, operators and generalizations)
Françoise Chaitin-Chatelin (numerical stability, book)
Brian Davies (pseudospectra of differential operators)
Nils Dencker (differential operators)
Mark Embree (matrix iterations, Toeplitz and random matrices)
Valerie Fraysse (numerical stability, parallel computation, book)
Stratis Gallopoulos (computation, iterative methods)
Sergei Godunov ("spectral portraits", numerical stability)
Nick Higham (Matlab software and examples, iterative computation, polynomial eigenvalue problems)
Diederich Hinrichsen (control, stability, "spectral value sets")
Yiannis Koutis (computation)
Shiu-Hong Lui (computation, pseudospectral mapping)
Michael Overton (optimization of pseudospectra)
Karel Pravda-Starov (differential operators)
Tony Pritchard (control, stability, "spectral value sets")
Johannes Sjöstrand (differential operators)
Françoise Tisseur (polynomial eigenvalue problems, 1-norm pseudospectra, control theory)
Nick Trefethen (theory, computation, applications)
Maciej Zworski (solvability of PDEs, pseudodifferential operators)

Others who have studied pseudospectra

Jeff Baggett (Hille-Phillips and Zabczyk operators)
Thierry Braconnier (iterative computation)
Martin Brühl (computation by curve-tracing)
Jean-Marc Chomaz (Ginzburg-Landau operator)
John Crawford (linearized Vlasov equation)
Jane Cullum (matrix iterations)
Jim Demmel (eigenvalue algorithms, first published computer-generated plot)
Jos van Dorsselaer (matrix pencils, stability of method of lines, general theory)
Toby Driscoll (wave operator with absorbing boundary)
Alan Edelman (matrix perturbations and normal forms)
Joe Flaherty (instability in ionosphere)
Eduardo Gallestey (computation, control theory)
Juan-Miguel Gracia (Jordan form, derogatory matrices)
Anne Greenbaum (matrix iterations, general theory)
Ali Harrabi (convergence of spectral approximations, pseudospectral definitions)
Dan Henningson (hydrodynamic stability and transition to turbulence)
Des Higham (numerical solution of ODEs)
Viktor Kostin ("spectral portraits", accuracy of algorithms)
Henry Landau (first to define "epsilon-spectra"?)
Ross Lippert (matrix perturbations and normal forms)
Andrew Lumsdaine (waveform relaxation, block Toeplitz operators)
Dany Mezher (parallel computation, visualization)
Brynjulf Owren (waveform relaxation, reaction-convection-diffusion equation)
Satish Reddy (fluids applications, various other contributions)
Lothar Reichel (Toeplitz matrices, matrix iterations)
Kurt Riedel (magnetohydrodynamics, generalized pseudospectra)
Steffen Roch (operator theory, Wiener-Hopf operators)
Axel Ruhe (rational Krylov algorithms)
Miloud Sadkane (computation, stability)
Peter Schmid (hydrodynamic stability and transition to turbulence)
Bernd Silbermann (Toeplitz matrices and operators, condition numbers)
Valeria Simoncini (computation, matrix iterations)
Kim-chuan Toh (companion matrices, matrix iterations, iterative computation)
Vincent Toumazou (computation, generalized pseudospectra, stability)
Anne Trefethen (parallel computation, fluids applications)
Andre Weideman (Papkovitch-Fadle operator)
Tom Wright (EigTool software, large-scale iterative computation, rectangular matrices)
Deyun Wu (waveform relaxation, block Toeplitz operators)

Others not detailed in the list above include Atallah, Bennani, Bernhardsson, Borba, Bulgakov, Cossu, Donato, Erhel, Giraud, Heuveline, Hochbruck, Huysmans, Jackiewicz, Kelb, Kerner, Knight, Kraaijevanger, Lubich, Malyshev, Marques, McCoy, Nachtigal, Pathria, Ottaviani, Plato, Razzakov, Renaut, Rigal, Spijker, Tal-Ezer, Trummer, Varah, Viswanath, Wolf.

Researchers involved in the more general study of non-normality include John Doyle (control), Brian Farrell and Petros Ioannou (atmospheric science, stability of fluid flow), Heinz-Otto Kreiss (stability of finite difference discretizations), David Nelson (non-Hermitian quantum mechanics), and Anthony Siegman (lasers).