Interpolating Some Classes of Operators between Families

Alexi Quevedo Suárez


Many operator ideals (single operator ideals and chains, see inside) possess the strong property of interpolation for the $J$ and $K$ methods of Lions-Peetre, Sparr, Fernandez and Cobos-Peetre. That is, let $\mathcal{I}$ be one of the ideals considered here, let $\bar{A}$ and $\bar{B}$ be interpolation families and $T : \bar{A}\to\bar{B}$ a bounded linear operator then, the interpolated operator $T_{J;K} : J(\bar{A})\to K(\bar{B})$ belongs to $\mathcal{I}$ if and only if the induced operator $T_{\mathcal{JS}}$ from the intersection space $\mathcal{J} (\bar{A})$ into the sum space $\mathcal{S}(B)$ belongs to $\mathcal{I}$.


Real methods of interpolation between families; Operator ideals

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B. Beauzamy, Banach-Saks Properties and Spreading Models, Math. Scand. 44(1979), 357-384.

J. Behrg and J. Lofstrom, Interpolation spaces. An introduction, Springer-Verlag, New York, 1976.

M.J. Carro, Real interpolation for families of Banach spaces, Studia Math. 109 (1994), 1-21.

M.J. Carro, Real interpolation for families of Banach spaces (II), Collect. Math. 45(1994), 53-83.

M.J. Carro and L.Y. Nikolova, Interpolation of limited and weakly compact operators on families of Banach spaces, Acta Appl. Math 49(1997), 151-177.

M.J. Carro, L.Y. Nikolova, J. Peetre and L.E. Persson, Some real interpolation methods for families of Banach spaces - A comparison, J. Approx. Theory 89(1997), 26-57.

F. Cobos and J. Peetre, Interpolation of compact operators: the multidimensional case, Proc. London Math. Soc. (3) 63(1991), 371-400.

R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher and G. Weiss, A theory of complex interpolation for families of Banach spaces, Adv. Math. 45(1982), 203-229.

W. Davis, T. Figiel, W. Johnson and A. Pelczynski, Factoring weakly compact operators, J. Funct. anal. 17(1974), 311-327.

D. L. Fernandez, Interpolation of $2^d$ Banach spaces and the Calderon spaces $mathcal{X}(E)$, Proc. London Math. Soc. (3) 56(1988), 143-162.

S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35(1980), 397-411.

K. Homan, Banach Spaces of analytic functions, Englewood Cliffs, Prentice-Hall, 1962.

A. Kryczka, Alternate signs Banach-Saks property and real interpolation of operators, Proc. Amer. Math. Soc., vol. 136, 10(2008), 3529-3537.

W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, in Handbook of the geometry of Banach spaces, North-Holland (2001).

J.L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math. 19(1964), 5-68.

J.R. Partington, Proc. Cambridge Philos. Soc. 82(1977), 369-374.

A. Pietsch, Operator ideals, North Holland, Amsterdam, 1980.

A. Quevedo, Factorization of mixed operators, Houston Journal of Mathematics, to appear. Available online at

G. Sparr, Interpolation of several Banach spaces, Ann. Mat. Pura Appl. 99(1974), 247-316.



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