## On the essential spectrum of matrix operators.(English. Russian original)Zbl 0871.47005

Math. Notes 58, No. 6, 1359-1362 (1995); translation from Mat. Zametki 58, No. 6, 945-949 (1995).
Let $$L=\left[\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right]$$ be an unbounded matrix operator acting on a Banach space. It is shown in the present paper that, under some suitable assumption, $$L$$ is closable if and only if $$S(\mu)$$ is and the essential spectrum of $$L$$ is the union of that of $$A$$ and $$S(\mu)$$.

### MSC:

 47A10 Spectrum, resolvent 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

### Keywords:

unbounded matrix operator; essential spectrum
Full Text:

### References:

 [1] I. A. Adam,Physics Reports (Review Section of Physics Letters),142, No. 5, 263–356 (1986). [2] S. Agmon, A. Douglis, and L. Nirenberg,Comm. Pure Appl. Math.,17, 35–92 (1964). · Zbl 0123.28706 [3] F. V. Atkinson, H. Langer, R. Mennicken, and A. A. Shkalikov,Math. Nachr.,167, 5–20 (1994). · Zbl 0831.47001 [4] I. P. Goedbloed, ”Lecture note on ideal magnetohydrodynamics,” in:Rijnhiuzen Report, Fom-Instituut voor Plasmafysica, Nieuwegein (1983), pp. 83–145. [5] G. Grubb and G. Geymonat,Math. Ann.,227, 247–276 (1977). · Zbl 0361.35050 [6] I. C. Gohberg, S. Goldberg, and M. A. Kaashoek,Classes of Operators, Vol. I, Oper. Theory: Advances and Appl.,49, Birkhauser-Verlag, Basel-Boston-Berlin (1990). · Zbl 0745.47002 [7] T. Kato,Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York (1980). · Zbl 0435.47001 [8] N. D. Kopachevskii, S. G. Krein, and Ngô Huy Can,Operator Methods in Linear Hydrodynamics. Evolution and Spectral Problems [in Russian], Nauka, Moscow (1989). [9] V. A. Malyshev and R. A. Minlos,Linear Operators in Infinite-Particle Systems [in Russian], Nauka, Moscow (1994). · Zbl 0839.46072 [10] M. A. Naimark,Linear Differential Operators [in Russian], Nauka, Moscow (1968). · Zbl 0227.34020 [11] A. E. Lifschitz,Magnetohydrodynamics and Spectral Theory, Kluwer Acad. Publishers, Dordrecht (1989). · Zbl 0698.76122
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.