Completeness of eigenvectors in Banach spaces

Benzinger, Harold E.

doi:10.1090/S0002-9939-1973-0318941-6

Completeness of eigenvectors in Banach spaces
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by Harold E. Benzinger PDF

Proc. Amer. Math. Soc. 38 (1973), 319-324 Request permission

Abstract:

We prove a general theorem on the completeness of the eigenvectors of linear operators in a Banach space. We then derive asymptotic estimates for the Green’s functions of two-point boundary value problems which allow us to apply the above theorem to a wide class of such problems in the spaces ${L^p}(0,1),1 \leqq p < \infty$.

References

Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
Harold E. Benzinger, Green’s function for ordinary differential operators, J. Differential Equations 7 (1970), 478–496. MR 262583, DOI 10.1016/0022-0396(70)90096-3
Harold E. Benzinger, The $L^{p}$ behavior of eigenfunction expansions, Trans. Amer. Math. Soc. 174 (1972), 333–344 (1973). MR 328189, DOI 10.1090/S0002-9947-1972-0328189-0
Israel Halperin, Closures and adjoints of linear differential operators, Ann. of Math. (2) 38 (1937), no. 4, 880–919. MR 1503380, DOI 10.2307/1968845
G. C. Rota, Extension theory of differential operators. I, Comm. Pure Appl. Math. 11 (1958), 23–65. MR 96852, DOI 10.1002/cpa.3160110103
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
J. Schwartz, Perturbations of spectral operators, and applications. I. Bounded perturbations, Pacific J. Math. 4 (1954), 415–458. MR 63568
A. S. Markus, Some tests of completeness of the system of root vectors of a linear operator and the summability of series in that system, Dokl. Akad. Nauk SSSR 155 (1964), 753–756 (Russian). MR 0162146
A. S. Markus, Certain criteria for the completeness of a system of root-vectors of a linear operator in a Banach space, Mat. Sb. (N.S.) 70 (112) (1966), 526–561 (Russian). MR 0216316