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(with the usual modification if <7 = 00) is finite.

If b = 0 we get the usual Besov spaces Bpg(R"). Besov spaces of generalized smoothness are Banach spaces and the norms are equivalent to each other for admissible choices of <p. These spaces attracted some attention in connection with fractal analysis and related spectral theory. A short description can be found in Triebel's book [30]. A detailed study of these spaces is made in the paper by Moura [23]. Let

with l<p<oo,l<Q<oo, and —00 < si < s0 < 00. Then Ao <-> j4i. Moreover (see [8] and [21])

B<^(R") = (B^(Rn),B^(Rn))geb;g where s = (1 — 9)so + Osi. Consequently, using Theorem 3.1 we derive:

Corollary 4.2. Let 0 < 6 < 1,1 < q < 00, b e R, and let 1 < p < 00, —00 < si < sq < 00 and s = (1 — 6)sq + 6s\. (i) Let b< 0. Then

where the infimum is taken over all representations oo

j=jo

The last application refers to operator spaces defined by summability conditions on the singular numbers. Let if be a Hilbert space and given any bounded linear operator T £ £(H), let {s„(T)} be the sequence of the singular numbers of T, defined by sn(T) = inf{||T — R\\ : rankiî < n} , n £ N.

For l<p<oo, l<ç<oo and 6 £ t, the Lorentz-Zygmund operator space CPtqtb{H) consists of all T £ C(H) having a finite norm oo n , , rP,g,b(T)=(E(ni~1( l + logny^CT))^-1) '

(with the usual modification if q = oo).

The space CPtgtb(H) is the component over H of the Lorentz-Zygmund operator ideal that has been studied in [5], [6] and [7]. Note that T belongs to CPiqib(H) if and only if {sn(T)} belongs to the Lorentz-Zygmund sequence space £p,q(log£)b- For b = 0 we recover the Lorentz operator space (Cpiq(H),Tptq), and for b = 0 and p = q,we get the Schatten p—class (.Cp(H),Tp) (see [16], [20] and [26]).

In order to apply Theorem 3.1 in this context, we put Aq = C\{H) and A\ = C(H). Then Aq c—> A\ and, according to [5, Thm. 5.1], we have

Theorem 3.1 yields:

Corollary 4.3. Let H be a Hilbert space. Let 1 < p < oo, 1 < q < oo, and let jo = jo(p) £ N such that for all j £ N with j > jo,

pUj p pV3 p

(i) Let b < 0. Then CPtq<b{H) is the set of all T £ C{H) such that oo \ «

(equivalent norms).

(ii) Let b > 0. Then Cp>qfi{H) consists of all T € C(H) which can be represented as T = Y^jLj0 Tj with Tj e Cp^j ¡q{H) such that i oo \ 9

Furthermore, the infimum over the expression in (9) is an equivalent norm in £Pigib(H).

Remark 4.1. All results we have shown refer to Banach spaces, but logarithmic interpolation spaces can also be considered in the class of quasi-Banach spaces. Details can be found in the joint paper by Fernández-Cabrera, Manzano, Martínez and the present author [9]. Among other things, we establish there that statement (ii) in Theorem 2.1 holds for 0 < p < 1 as well, and we give applications to operator spaces on Banach spaces.

References

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