For the classical Hardy operator $T\colon \ell^p\to \ell^p \quad (Tx)_n=\frac{1}{n}\sum_{k=1}^n x_k$ or the integral type $S\colon L^p\rightarrow L^p \quad (Sf)(x)=\frac{1}{x}\int_0^x f(t) dt \ \ $ the norm is well known to be $\frac{p}{p-1}$ for $1 . I did some research but did not find the result for the adjoint operator $T'\colon \ell^p\rightarrow \ell^p \quad (T'x)_n=\sum_{k=n}^{\infty} \frac{x_k}{k}$ or its integral version. What does adjoint mean in the case of general Banachspaces at all? For $p=2$ it's easy to verify $\langle Hx,y\rangle =\langle x,H'y\rangle$.
Norm of conjugate Hardy operator
1
$\begingroup$
functional-analysis
operator-theory
adjoint-operators