By Dr. rer. nat. Albrecht Böttcher, Prof. Dr. sc. nat. Bernd Silbermann (auth.)

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**Extra resources for Analysis of Toeplitz Operators**

**Example text**

Inf HrpJloo (where,in fact, the inf can be replaced by min). ButOis an extension of Cif and onlyif J(rp - AXo) h dm = 0 V hEHl T § (rp - AXo)n = 0 § (Xlrp)n - an § Xlrp - b E Hoo , = 0 Vn ~ 0 Vn ~ 1 therefore, <1>(A) = inf {JlrpJloo: Xlrp - b E HOO} = inf {llxwJloo: Xlrp - b E HOO} and now (2) gives (1). = dist Loo (b, HOO) • Remark. Let the hypotheses of the preceding theorem be satisfied. Then AXo E HP and so there exists a function a E HP whose n-~h Fourier coefficient is an (n E Z+). One can now formulate the following criterion for the boundedness of A: A E I(HP) if and only if a E BMO.

Jq,(f} 27t '1') l(e1'1') dtp, o where q,(f}) = 2r sin 1J/(1 - 2r cos1J + r 2 ). P. Using the Riesz projeetion one ean write 1= -i(PI- QI-/o) = -i(2PI-I-/o)· (1) 39 LP and HP We therefore deduce from the M. 41 that sup {JilJlp/Jl/Jlp: I E (P, 1=1= O} is finite for 1 < P < 00 and infinite for p = 1 and p = 00. Thus, the operator of conjugation, 11-+ 1, is bounded on Lp for 1 < p < 00. In particular, (1) makes a correct sense for arbitrary I E Lp (1 < P < (0). The conjugation operator 11-+ is frequently also referred to as the Hilbert translorm.

9. Shift operators. Reca11 that Xn is given by Xn(t) = ,n (I E T). The operators M(XI) and T(XI) are usua11y referred to as the bilateral and unilateral 8ki/t, respectively, because they act on IP(Z) resp. •}. Also as usual, we put U:= M(XI) snd V:= T(Xl), and, for nE Z+, we let Un = M(Xn) , u-n = M(X-n) , vn = T(Xn) , V(-n) = T(X-n). 53 Hankeloperators Then obviously U±" = (U±l)", V" = (VI)", VI-li) = (VI-I)", and u-nun = unu- n = I, U* = U-l, VI-")V" = I, V* = VI-I). Note that vn VI-n) is not the identity operator.