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Conditional multipliers and essential norm of uC φ between L p spaces. (English) Zbl 1200.47029

Let (X,Σ,μ) be a σ-finite measure space. The space of Σ-measurable functions on X is denoted by L 0 (Σ), and if p<, then L p (Σ) denotes {[f]:[f] p =p p <}, where [f] is the equivalence class of functions which differ from f on sets of measure 0, and for 1p, · p is the L p -norm.

In the paper, 𝒜Σ denotes a complete σ-finite sub-algebra of Σ, and the operator E 𝒜 =E is said to be a conditional expectation operator if for f0, fL p (Σ), E(f) is 𝒜-measurable and 𝒜 E(f)dμ= 𝒜 fdμ, whenever the integrals are finite. If 1p,q<, then the set of conditional multipliers K p,q =K p,q (𝒜,Σ) is defined to be {uL 0 (Σ):uL p (𝒜)L q (Σ)}, M u :L p (𝒜)L q (Σ) denotes the multiplication operator M u (f)=uf, and if φ:XX is a non-singular measurable transformation with measure uφ -1 which is absolutely continuous with respect to μ, the derivative d(pφ -1 )/dμ is denoted by h. If C φ denotes the composition operator, C φ (f)(x)=fφ(x)=f(φ(x)), then K p,q φ is defined to be {uL 0 (Σ):uRange(C φ )L q (Σ)}, and

uC φ f q q = X S|f| q dμ,

1q<, where S=hE φ -1 (Σ)/(E 𝒜 (|u| q ))φ -1 . The results of this paper include:

(K p,q ,· p,q ) is a Banach space, where u p,q =E(|u| q ) 1/q r , if 1qp and 1/r=(1/q)-(1/p), and u p,q ={sup n E(|u| q )(A n )/μ(A n ) 1/r }, if 1pq, 1/r=(1/p)-(1/q), and {A n } is a countable set of pairwise disjoint 𝒜-atoms such that X=B(A n );

characterizations indicating that uK p,q φ if and only if SL 1/(1-(q(p)) (Σ), where 1qp; uK p,q φ if and only S=0 a.e on B and sup n {S(A n )/μ(A n ) (q(p))-1 }<, where 1pq.

MSC:
47B20Subnormal operators, hyponormal operators, etc.
47B38Operators on function spaces (general)