The essential norm of an operator and its adjoint

Axler, Sheldon; Jewell, Nicholas; Shields, Allen

doi:10.1090/S0002-9947-1980-0576869-9

The essential norm of an operator and its adjoint
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by Sheldon Axler, Nicholas Jewell and Allen Shields PDF

Trans. Amer. Math. Soc. 261 (1980), 159-167 Request permission

Abstract:

We consider the relationship between the essential norm of an operator T on a Banach space X and the essential norm of its adjoint $T^{\ast }$. We show that these two quantities are not necessarily equal but that they are equivalent if $X^{\ast }$ has the bounded approximation property. For an operator into the sequence space ${c_0}$, we give a formula for the distance to the compact operators and show that this distance is attained. We introduce a property of a Banach space which is useful in showing that operators have closest compact approximants and investigate which Banach spaces have this property.

References

Erik M. Alfsen and Edward G. Effros, Structure in real Banach spaces. I, II, Ann. of Math. (2) 96 (1972), 98–128; ibid. (2) 96 (1972), 129–173. MR 352946, DOI 10.2307/1970895
Sheldon Axler, I. David Berg, Nicholas Jewell, and Allen Shields, Approximation by compact operators and the space $H^{\infty }+C$, Ann. of Math. (2) 109 (1979), no. 3, 601–612. MR 534765, DOI 10.2307/1971228
Arlen Brown and Carl Pearcy, Introduction to operator theory. I, Graduate Texts in Mathematics, No. 55, Springer-Verlag, New York-Heidelberg, 1977. Elements of functional analysis. MR 0511596

Linear operators

On a certain subset of

and nonoperators

On subspaces of spaces with unconditional basis and spaces of operators

Julien Hennefeld, A decomposition for $B(X)^{\ast }$ and unique Hahn-Banach extensions, Pacific J. Math. 46 (1973), 197–199. MR 370265
Richard Holmes, Bruce Scranton, and Joseph Ward, Approximation from the space of compact operators and other $M$-ideals, Duke Math. J. 42 (1975), 259–269. MR 394301
W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
Ka Sing Lau, On a sufficient condition for proximity, Trans. Amer. Math. Soc. 251 (1979), 343–356. MR 531983, DOI 10.1090/S0002-9947-1979-0531983-0
Åsvald Lima, $M$-ideals of compact operators in classical Banach spaces, Math. Scand. 44 (1979), no. 1, 207–217. MR 544588, DOI 10.7146/math.scand.a-11804
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
Daniel H. Luecking, The compact Hankel operators form an $M$-ideal in the space of Hankel operators, Proc. Amer. Math. Soc. 79 (1980), no. 2, 222–224. MR 565343, DOI 10.1090/S0002-9939-1980-0565343-7
Harold S. Shapiro, Fourier multipliers whose multiplier norm is an attained value, Linear operators and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1971) Internat. Ser. Numer. Math., Vol. 20, Birkhäuser, Basel, 1972, pp. 338–347. MR 0390623
R. R. Smith and J. D. Ward, $M$-ideal structure in Banach algebras, J. Functional Analysis 27 (1978), no. 3, 337–349. MR 0467316, DOI 10.1016/0022-1236(78)90012-5