---
title: 'Rhaly Operator: Terraced Summability in Analysis'
url: https://www.emergentmind.com/topics/rhaly-operator
type: topic
---

# Rhaly Operator: Terraced Summability in Analysis

The Rhaly operator is a lower-triangular terraced operator determined by a scalar sequence and defined by multiplying each initial partial sum of a sequence, or of Taylor coefficients, by a row-dependent weight. In its basic one-variable form, if \((a_n)\) or \((\eta_n)\) is a complex sequence, then the corresponding matrix has constant entries along each row up to the diagonal, and the induced action is
\[
(R_a x)_n=a_n\sum_{k=0}^n x_k
\qquad\text{or}\qquad
(\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n
\]
for \(f(z)=\sum_{k\ge 0} a_k z^k\). The special case \(a_n=\eta_n=1/(n+1)\) is the classical Cesàro operator. Recent work places Rhaly operators at the intersection of summability theory, Hardy–Bergman analysis, factorable matrices, weighted sequence spaces, Köthe spaces, and several-complex-variable operator theory [2511.09201], [2503.21930], [2507.20214], [2606.15875], [2606.25885].

## 1. Definition, matrix structure, and basic identities

A Rhaly matrix is lower triangular and terraced: the \(n\)th row is constant up to the main diagonal. In the \(0\)-indexed convention used for Hardy-space and \(\ell^2\) formulations, its entries are
\[
R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),
\]
so that
\[
(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).
\]
In the \(1\)-indexed convention used on weighted null sequence spaces and Köthe spaces,
\[
a_{n,k}=a_n \quad (1\le k\le n),\qquad a_{n,k}=0\quad (k>n),
\]
and
\[
(R_a x)_n=a_n\sum_{k=1}^n x_k.
\]
The two conventions are notational variants of the same terraced construction [2503.21930], [2510.10550], [2507.20214].

For analytic functions, the coefficient action is central. If
\[
f(z)=\sum_{k=0}^\infty a_k z^k,\qquad F_{(\eta)}(z)=\sum_{n=0}^\infty \eta_n z^n,
\]
then
\[
\mathcal{R}_{(\eta)} f(z)
= \sum_{n=0}^\infty \Big(\eta_n\sum_{k=0}^n a_k\Big) z^n
= F_{(\eta)}(z)\star \Big(\frac{f(z)}{1-z}\Big),
\]
where \(\star\) denotes Hadamard convolution,
\[
(g\star h)(z)=\sum_{n=0}^\infty \big(\widehat g(n)\,\widehat h(n)\big) z^n.
\]
This identity is one of the defining analytic features of the operator [2511.09201].

Several equivalent operator factorizations recur throughout the literature. On \(\ell^2\),
\[
R_\alpha=D_\alpha H,
\]
where \(H\) is the discrete Hardy operator \((Hx)_n=\sum_{k=0}^n x_k\) and \(D_\alpha\) is diagonal, \((D_\alpha x)_n=\alpha_n x_n\) [2503.21930]. In Cesàro-based formulations,
\[
R_a=D_a C,
\]
with \(C\) the classical Cesàro matrix and \(D_a=\operatorname{diag}((n+1)a_n)\) [2408.03182]. On weighted Hardy spaces \(H^2(\omega)\), the operator becomes a factorable matrix \(F_{\alpha,\beta}\) in the orthonormal basis \(\phi_n(z)=z^n/\sqrt{\omega_n}\), with
\[
\alpha_n=\eta_n\sqrt{\omega_n},\qquad \beta_n=\frac{1}{\sqrt{\omega_n}},\qquad r_{k,j}=\alpha_k\beta_j \ \ (j\le k),
\]
connecting Rhaly theory directly to the theory of factorable matrices [2606.15875].

A persistent misconception is to identify the Rhaly operator with a Toeplitz, Nörlund, or Cauchy-product convolution operator. The coefficient rule
\[
b_n=\eta_n\sum_{k=0}^n a_k
\]
is not of the form \(b_n=\sum_{k=0}^n \eta_{n-k}a_k\), so the matrix is lower triangular but non-Toeplitz. In the Hardy-space setting it is also distinct from multiplication by the generating function \(F_{(\eta)}\); boundedness of the multiplier \(M_G:f\mapsto Gf\) for \(G\in H^\infty\) does not imply boundedness of \(\mathcal R_{(\eta)}\) [2511.09201].

## 2. Hardy, Bergman, and weighted Hardy spaces

The paper "Rhaly operators acting on Hardy and Bergman spaces" identifies mean Lipschitz regularity of the generating function \(F_{(\eta)}\) as the decisive criterion for boundedness and compactness on classical analytic function spaces [2511.09201]. For \(1\le p\le\infty\) and \(0<\alpha\le 1\), the mean Lipschitz space \(\Lambda^p_\alpha\) is characterized by
\[
\Lambda^p_\alpha
=
\Big\{ f:\ M_p(r,f')=O\big((1-r)^{\alpha-1}\big)\text{ as }r\to 1^- \Big\},
\]
and its little-oh variant \(\lambda^p_\alpha\) requires
\[
M_p(r,f')=o\big((1-r)^{\alpha-1}\big).
\]

On Hardy spaces \(H^p\), the main dichotomy is at \(p=2\). If \(1<p\le 2\), then
\[
\mathcal R_{(\eta)}\in \mathcal B(H^p)
\iff
F_{(\eta)}\in \Lambda^p_{1/p},
\]
and
\[
\mathcal R_{(\eta)}\in \mathcal K(H^p)
\iff
F_{(\eta)}\in \lambda^p_{1/p}.
\]
If \(2<p<\infty\), boundedness implies \(F_{(\eta)}\in \Lambda^p_{1/p}\), and compactness implies \(F_{(\eta)}\in \lambda^p_{1/p}\), but sufficiency is presently established only under the stronger hypotheses
\[
F_{(\eta)}\in \Lambda^q_{1/q}\quad\text{or}\quad F_{(\eta)}\in \lambda^q_{1/q}
\]
for some \(2<q<p\) [2511.09201]. The same work states explicitly that a full if-and-only-if characterization for \(p>2\) remains open.

For unweighted Bergman spaces \(A^p\), the same paper obtains exact criteria for all \(1<p<\infty\):
\[
\mathcal R_{(\eta)}\in \mathcal B(A^p)\iff F_{(\eta)}\in \Lambda^p_{1/p},
\qquad
\mathcal R_{(\eta)}\in \mathcal K(A^p)\iff F_{(\eta)}\in \lambda^p_{1/p}.
\]
More generally, on weighted Bergman spaces \(A^p_\alpha\) with \(\alpha>-1\), mean Lipschitz conditions suffice for boundedness and compactness, and in the range \(-1<\alpha<2p-2\) they are also necessary [2511.09201].

The \(\ell^2\)-based Hardy theory is sharper in Schatten-class terms. For the one-variable Rhaly matrix \(R_\alpha\) on \(\ell^2\), which is identified with \(H^2\), boundedness is equivalent to each of the following:
\[
f_\alpha(z)=\sum_{n\ge 0}\alpha_n z^n\in \Lambda^2_{1/2},
\]
\[
\sup_{n\ge 0}2^n\sum_{j=2^n}^{2^{n+1}-1}|\alpha_j|^2<\infty,
\]
and
\[
\sup_{k\ge -1}\sigma_k<\infty,
\qquad
\sigma_k=\Big(\sum_{j=2^k}^{2^{k+1}-1}(j+1)|\alpha_j|^2\Big)^{1/2}.
\]
Compactness is equivalent to the corresponding vanishing dyadic condition, or equivalently to \(f_\alpha\in \lambda^2_{1/2}\). For \(1<q<\infty\),
\[
R_\alpha\in \mathcal S^q(\ell^2)
\iff
\sum_{n=0}^\infty
\left(
2^n\sum_{j=2^n}^{2^{n+1}-1}|\alpha_j|^2
\right)^{q/2}<\infty,
\]
equivalently \(f_\alpha\in B^{2,q}_{1/2}\). In particular,
\[
R_\alpha\in \mathcal S^2
\iff
\sum_{n=0}^\infty (n+1)|\alpha_n|^2<\infty
\]
[2503.21930].

On weighted Hardy spaces \(H^2(\omega)\), boundedness and compactness are governed by factorable-matrix parameters. Writing
\[
K_2:=\sup_{m\ge 0}
\Big(\sum_{k=m}^\infty |\eta_k|^2\omega_k\Big)
\Big(\sum_{j=0}^m\omega_j^{-1}\Big),
\]
and
\[
K_1:=
\sup_{m\ge 0}
\Big(\sum_{j=0}^m\omega_j^{-1}\Big)^{-1}
\sum_{k=0}^m
|\eta_k|^2\omega_k
\Big(\sum_{j=0}^k\omega_j^{-1}\Big)^2,
\]
one has
\[
R_\eta\text{ bounded on }H^2(\omega)
\iff
K_1<\infty
\iff
K_2<\infty,
\]
with
\[
K_1\le 2K_2^2\le 2\|R_\eta\|_{B(H^2(\omega))}^2\le 8K_1.
\]
Compactness is characterized by the corresponding little-oh conditions [2606.15875].

This weighted theory recovers familiar spaces. For \(A^2_\gamma\), \(\gamma>-1\), boundedness is equivalent to
\[
\sum_{k=0}^m |\eta_k|^2 (k+1)^{\gamma+3}=O(m^{\gamma+2}),
\]
which is equivalent to
\[
\sum_{k\ge m} |\eta_k|^2=O(1/m),
\]
and therefore to \(f_\eta\in \Lambda(2,1/2)\). For \(\mathcal D_\alpha\), \(0<\alpha\le 1\), boundedness is again equivalent to the same tail condition. For the critical Dirichlet space \(\mathcal D=\mathcal D_0\), the criterion changes to
\[
\sum_{k\ge m} k|\eta_k|^2=O(1/\log m),
\]
and for Schatten class \(\mathcal S^2\) an additional logarithmic factor appears:
\[
R_\eta\in \mathcal S^2(\mathcal D)
\iff
\sum_{k=0}^\infty (k+1)|\eta_k|^2\log k<\infty.
\]
By contrast, on \(H^2\), \(A^2_\gamma\), and \(\mathcal D_\alpha\) with \(0<\alpha\le 1\), the dyadic condition
\[
\sum_{k=0}^\infty
\left(
2^k\sum_{j=2^k}^{2^{k+1}-1}|\eta_j|^2
\right)^{p/2}<\infty
\]
characterizes \(\mathcal S^p\)-membership for \(1<p<\infty\), independently of \(\gamma\) and of \(\alpha\in(0,1]\) [2606.15875].

The classical Cesàro operator sits exactly at the threshold between boundedness and compactness in many of these settings. For \(\eta_n=1/(n+1)\), the operator is bounded on \(H^p\) and \(A^p_\alpha\), but on \(\ell^2\) it is neither compact nor Schatten-class. The paper on Hardy and Bergman spaces further isolates the borderline \(O(1/n)\) decay: if \(2\le p<\infty\) and \(\eta_n=O(1/n)\), then \(\mathcal R_{(\eta)}\) is bounded on \(H^p\); however, there exists a sequence with \(\eta_n=O(1/n)\) for which \(\mathcal R_{(\eta)}\) is not bounded on \(H^p\) when \(1\le p<2\). For decreasing nonnegative \((\eta_n)\), boundedness on \(H^p\), \(1<p<\infty\), is equivalent to \(\eta_n=O(1/n)\) [2511.09201].

## 3. Weighted null sequence spaces and fine spectral structure

A second major line of work studies Rhaly operators on weighted null sequence spaces
\[
c_0(r)=\{x=\{x_k\}\in \mathbb C^{\mathbb N}: \lim_{k\to\infty} r_k x_k=0\},
\qquad
\|x\|_r=\sup_k |x_k|r_k,
\]
where \(r=(r_n)\) is strictly positive. If \(\inf_k r_k>0\), then \(c_0(r)=c_0\) with equivalent norms; the genuinely weighted regime is \(\inf_k r_k=0\) [2510.10550], [2305.04624].

For bounded, strictly positive weights \(r,s\), the exact boundedness criterion for
\[
R_a:c_0(r)\to c_0(s),\qquad (R_a x)_n=a_n\sum_{k=1}^n x_k,
\]
is
\[
\left\{s_n a_n\sum_{k=1}^n \frac{1}{r_k}\right\}_{n\in\mathbb N}\in \ell_\infty,
\]
and then
\[
\|R_a\|_{r,s}
=
\sup_{n\in\mathbb N}
s_n a_n\sum_{k=1}^n \frac{1}{r_k}.
\]
Compactness is characterized by
\[
s_n a_n\sum_{k=1}^n \frac{1}{r_k}\to 0.
\]
In the equal-weight case \(r=s\), this is often written using
\[
b_n:=a_n s_n\sum_{k=1}^n \frac{1}{s_k},
\]
with \(R_a\in B(c_0(s))\iff (b_n)\in \ell_\infty\) and \(R_a\in K(c_0(s))\iff b_n\to 0\) [2305.04624], [2510.10550].

Several sufficient conditions are weight-driven rather than sequence-driven. If \(s\) is strictly positive and
\[
0\le \limsup_{n\to\infty}\frac{s_{n+1}}{s_n}<1,
\]
then bounded \((a_n)\) implies boundedness of \(R_a:c_0(s)\to c_0(s)\), and \(a_n\to 0\) implies compactness. Geometric weights \(s_n=q^n\), \(0<q<1\), fall directly into this regime. The literature also emphasizes that this decay condition is sufficient, not necessary: for example, with
\[
a_n=\frac{1}{n^p},\qquad s_n=e^{-n^{1/\alpha}},
\]
where \(p>1\) and \(\alpha\in\mathbb N\setminus\{1\}\), one still has
\[
\lim_{n\to\infty} a_n s_n\sum_{k=1}^n \frac1{s_k}=0,
\]
hence compactness, even though \(\lim s_{n+1}/s_n=1\) and the simple sufficient condition does not apply [2510.10550].

The compact case admits a complete spectral description under standard monotonicity and positivity hypotheses. Assume \(a=(a_n)\) is a sequence of non-zero real numbers, \(s\) is a bounded decreasing sequence of strictly positive reals, and \(R_a\in K(c_0(s))\). Writing
\[
S:=\{a_n:n\in\mathbb N\},
\]
the spectrum satisfies
\[
\sigma_p(R_a,c_0(s))=S,\qquad
\sigma(R_a,c_0(s))=S\cup\{0\},
\]
\[
\sigma_c(R_a,c_0(s))=\{0\},\qquad
\sigma_r(R_a,c_0(s))=\varnothing.
\]
Moreover,
\[
\sigma_{ap}(R_a,c_0(s))=S\cup\{0\},\qquad
\sigma_\delta(R_a,c_0(s))=S\cup\{0\},\qquad
\sigma_{co}(R_a,c_0(s))=S,
\]
and the spectral radius is
\[
r(R_a)=\sup_n |a_n|.
\]
Goldberg’s classification places all nonzero spectral points in class \(III_3\) and \(0\) in class \(II_2\) [2510.10550].

A related but distinct weighted-\(c_0\) spectral theory arises when the asymptotic parameter
\[
\chi=\lim_{n\to\infty} n a_n
\]
exists and is nonzero. Under this assumption, the paper "Spectral properties of the Rhaly operator on weighted null sequence spaces and associated operator ideals" gives product-estimate descriptions of point and residual spectra. If
\[
S:=\{a_n:n\in\mathbb N\},\qquad \alpha=\Re(1/\lambda),
\]
then
\[
\sigma_p(R_a,c_0(s))
=
\left\{
\lambda\in S:\ \lim_{n\to\infty} a_n s_n\, n^{\alpha\chi}=0
\right\},
\]
while
\[
\sigma_p(R_a^*,c_0(s)^*)
\supseteq S
\]
and, for \(\lambda\notin \overline S\),
\[
\lambda\in \sigma_p(R_a^*,c_0(s)^*)
\iff
\sum_{n=1}^\infty \frac{1}{s_n n^{\alpha\chi}}<\infty.
\]
In this regime the spectrum outside \(\overline S\) is contained in the closed disk centered at \(\chi/2\) with radius \(\chi/2\), yielding the bound \(r(R_a)\le \chi\) [2305.04624].

These weighted-sequence-space results show that terraced lower-triangular structure can lead either to fully discrete compact spectra \(S\cup\{0\}\), or to Cesàro-type disk localization, depending on the asymptotic regime imposed on the terrace sequence.

## 4. Köthe spaces, power series spaces, and operator dynamics

Rhaly operators extend naturally to Köthe sequence spaces. If \((a_{n,k})_{n,k\in\mathbb N}\) is a Köthe matrix, the Köthe space is
\[
K(a_{n,k})
=
\left\{
x=(x_n)_{n\in\mathbb N}:
\|x\|_k:=\sum_{n=1}^\infty |x_n|a_{n,k}<\infty
\text{ for all }k
\right\},
\]
with dual
\[
K(a_{n,k})'
=
\left\{
y=(y_n):
\sup_n |y_n|a_{n,k}^{-1}<\infty
\text{ for some }k
\right\}.
\]
For a terraced sequence \(\theta=(\theta_n)\), the associated operator satisfies
\[
R_\theta e_n=\sum_{j=n}^\infty \theta_j e_j,
\qquad
R_\theta x=\sum_{n=1}^\infty \Big(\sum_{j=1}^n x_j\Big)\theta_n e_n.
\]
A general continuity criterion states that
\[
R_\theta:K(a_{n,k})\to K(b_{n,k})
\]
is well defined and continuous if and only if \(\theta\in K(b_{n,k})\) and, for every \(k\), there exists \(m\) such that
\[
\sup_{n\in\mathbb N}\frac{\|R_\theta e_n\|_k}{\|e_n\|_m}<\infty.
\]
If \(\theta\in K(b_{n,k})\) and \(\inf_n a_{n,m_0}>0\) for some \(m_0\), continuity follows; if moreover \(K(b_{n,k})\) is Montel, then \(R_\theta\) is compact [2507.20214].

The paper "The Rhaly Operators on Köthe Spaces" specializes this theory to power series spaces. For an increasing sequence \(\alpha_n\to\infty\),
\[
\Lambda_1(\alpha)
=
\left\{
x=(x_n):
\|x\|_k=\sum_{n=1}^\infty |x_n|e^{-\alpha_n/k}<\infty \text{ for all }k
\right\},
\]
and
\[
\Lambda_\infty(\alpha)
=
\left\{
x=(x_n):
\|x\|_k=\sum_{n=1}^\infty |x_n|e^{k\alpha_n}<\infty \text{ for all }k
\right\}.
\]
The standard identifications
\[
H(\mathbb C)\cong \Lambda_\infty(n),
\qquad
H(\mathbb D)\cong \Lambda_1(n)
\]
place Rhaly operators into holomorphic function theory [2507.20214].

On infinite-type spaces, the behavior is especially rigid. If \(\theta\in \Lambda_\infty(\beta)\), then
\[
R_\theta:\Lambda_\infty(\alpha)\to \Lambda_\infty(\beta)
\]
is continuous and compact; likewise, if \(\theta\in \Lambda_1(\beta)\), then
\[
R_\theta:\Lambda_\infty(\alpha)\to \Lambda_1(\beta)
\]
is continuous and compact. On finite-type spaces, continuity from \(\Lambda_1(\alpha)\) to \(\Lambda_1(\beta)\) follows under the comparison condition
\[
\alpha_n\le A\beta_n+B,
\]
and for a nuclear \(\Lambda_1(\alpha)\) one has the sharp self-map criterion
\[
R_\theta:\Lambda_1(\alpha)\to \Lambda_1(\alpha)\text{ compact}
\iff
\theta\in (\Lambda_1(\alpha))'.
\]
In particular, the classical Cesàro operator is not compact on \(\Lambda_1(\alpha)\) because the sequence \((1/(n+1))\) does not belong to the strong dual \((\Lambda_1(\alpha))'\) [2507.20214].

The same paper develops topological and ergodic properties. If \(\inf_n a_{n,m_0}>0\) for some \(m_0\), then \(R_\theta\) is \(m\)-topologizable on \(K(a_{n,k})\). On \(\Lambda_\infty(\alpha)\), every \(R_\theta\) with \(\theta\in\Lambda_\infty(\alpha)\) is \(m\)-topologizable and power bounded. On nuclear \(\Lambda_1(\alpha)\), power boundedness is characterized by
\[
\sup_{p\in\mathbb N}\|\theta\|_p\le 1.
\]
Using the ABR and KT criteria cited in that paper, one obtains the equivalence, on nuclear \(\Lambda_1(\alpha)\), between power boundedness, mean ergodicity, uniform mean ergodicity, and Cesàro boundedness together with
\[
\lim_{n\to\infty}\frac{R_\theta^n}{n}=0
\]
pointwise [2507.20214].

Integral representations tie this locally convex theory back to the terraced matrix. For \(g\in H(\mathbb C)\),
\[
(R_g f)(z)
=
\frac{1}{2\pi i}\oint_{|w|=r_0}
\frac{f(w)}{w(1-w)}\,g(z/w)\,dw,
\qquad 0<r_0<1,
\]
and the associated terrace sequence is given by the Taylor coefficients of \(g\). An analogous formula holds on \(H(\mathbb D)\). The paper deduces from this that on \(H(\mathbb C)\), \(R_g\) is compact for every \(g\in H(\mathbb C)\), while on \(H(\mathbb D)\) compactness is equivalent to \(g\) belonging to the strong dual of \(H(\mathbb D)\) [2507.20214].

## 5. Generalized Cesàro operators and related Rhaly families

A closely related family is the generalized Cesàro, or Rhaly, operator
\[
(C_t x)_n=\frac{1}{n+1}\sum_{k=0}^n t^{\,n-k}x_k,
\qquad t\in[0,1),
\]
with matrix entries
\[
a_{n,k}(t)=\frac{t^{\,n-k}}{n+1}\quad (0\le k\le n).
\]
At \(t=1\) this reduces to the classical Cesàro operator. For \(t<1\), the subdiagonals are geometrically tilted [2302.08750].

On a broad class of translation-invariant solid Banach lattices with natural basis, including all separable rearrangement-invariant sequence spaces, various weighted \(\ell^p(w)\) and \(c_0(w)\) spaces, Cesàro spaces \(\mathrm{ces}_p\), dual-type spaces \(d_p\), and Bachelis spaces \(N^p\), one has a uniform compactness phenomenon:
\[
C_t=D\,R_t,\qquad
R_t=\sum_{m=0}^\infty t^m S^m,
\]
where \(D\) is the diagonal operator \(Dx=((x_n/(n+1))_n)\), and \(D\) is compact while \(R_t\) is bounded. Hence \(C_t\) is compact for every \(t\in[0,1)\). Its spectrum is fully explicit:
\[
\sigma(C_t;X)=\left\{\frac1{n+1}:n\in\mathbb N_0\right\}\cup\{0\},
\]
\[
\sigma_p(C_t;X)=\left\{\frac1{n+1}:n\in\mathbb N_0\right\},
\qquad
\sigma_c(C_t;X)=\{0\},
\qquad
\sigma_r(C_t;X)=\varnothing
\]
whenever the canonical basis is a basis. In the non-separable Cesàro space \(\mathrm{ces}_\infty\), by contrast, \(0\) lies in the residual spectrum [2302.08750].

The explicit eigenvectors are given, for each \(m\in\mathbb N_0\), by a sequence \(x^{[m]}\) with \(x_k^{[m]}=0\) for \(k<m\), \(x_m^{[m]}=1\), and
\[
\frac{x_{m+n+1}^{[m]}}{x_{m+n}^{[m]}}
=
\frac{(m+n+1)t}{n+1},
\]
so the ratio tends to \(t<1\) and the vector belongs to the decreasing-majorant space \(d_1\), hence to all spaces under consideration [2302.08750].

A weighted-\(c_0\) analogue yields the same spectral pattern in the compact case. For compact generalized Cesàro operators
\[
(C_t x)_n=\frac1n \sum_{k=1}^n t^{n-k}x_k
\]
on \(c_0(r)\to c_0(s)\), boundedness and compactness are characterized by the sequence
\[
\mu_n=\frac{s_n}{n}\sum_{k=1}^n \frac{t^{n-k}}{r_k},
\]
with boundedness equivalent to \((\mu_n)\in \ell^\infty\) and compactness equivalent to \((\mu_n)\in c_0\). In the compact case,
\[
\sigma_p(C_t,c_0(s))=\{1/n:n\in\mathbb N\},\qquad
\sigma(C_t,c_0(s))=\{1/n:n\in\mathbb N\}\cup\{0\},
\]
\[
\sigma_c(C_t,c_0(s))=\{0\},\qquad
\sigma_r(C_t,c_0(s))=\varnothing,
\]
with the same Goldberg classification pattern as for compact Rhaly operators [2510.10550].

This generalized Cesàro theory is structurally adjacent to, but not identical with, the terraced Rhaly operator. It retains the lower-triangular initial-sum architecture but inserts a geometric kernel \(t^{n-k}\), yielding a family that is compact for \(t<1\) in many spaces where the classical Cesàro operator \(C_1\) is not compact.

## 6. Moment representations, invariant subspaces, several variables, and open directions

When the terrace sequence is a moment sequence
\[
a_n=\mu_n=\int_{[0,1)} t^n\,d\mu(t),
\]
the Rhaly operator admits an integral representation on \(H^2\):
\[
R_\mu f(z)
=
\int_{[0,1)} \frac{f(tz)}{1-tz}\,d\mu(t).
\]
Equivalently,
\[
R_\mu f(z)=\int_{[0,1)} W_t f(z)\,d\mu(t),
\qquad
W_t f(z)=\frac{f(tz)}{1-tz}.
\]
This formulation supports spectral and invariant-subspace results unavailable for a general terrace sequence [2408.03182].

For bounded \(R_\mu\) on \(\ell^2\), letting
\[
s_n=\sum_{j=0}^n \mu_j,
\]
the point spectrum is characterized by
\[
\mu_k\in \sigma_p(R_\mu)
\iff
(\mu_n\exp(s_n/\mu_k))_{n\ge 0}\in \ell^2.
\]
Consequently,
\[
\sigma_p(R_\mu)
=
\left\{
\mu_k:
(\mu_n\exp(s_n/\mu_k))_{n\ge 0}\in \ell^2
\right\}.
\]
This recovers the classical Cesàro fact \(\sigma_p(C)=\varnothing\) for Lebesgue measure, since then \(\mu_n=1/(n+1)\) and \(s_n\sim \log n\); it also yields \(\sigma_p(R_{\delta_t})=\{t^k:k\ge 0\}\) for a point mass \(\delta_t\), \(0<t<1\) [2408.03182].

The adjoint can be substantially larger spectrally. If \(\lambda\in \sigma_p(R_\mu^*)\), then \(\lambda\neq 0\). Moreover, if \(\gamma>0\) satisfies
\[
\exp(-(1+\varepsilon)\gamma s_n)\in \ell^1
\quad\text{for all }\varepsilon>0,
\]
then the open disk
\[
\{z\in \mathbb C: |z-1/\gamma|<1/\gamma\}
\subset \sigma_p(R_\mu^*).
\]
For measures with density \(t^\alpha\), \(\alpha>-1\), this reproduces the Cesàro-type disk
\[
\{z:|z-1|<1\}\subset \sigma_p(R_\mu^*).
\]
The same paper also proves a numerical-range localization:
\[
W(R_\mu)\subset \{z\in\mathbb C:\Re z\ge 0\}=: \mathbb C_+,
\qquad
\sigma(R_\mu)\subset \mathbb C_+,
\]
and deduces that \(-R_\mu\) is the infinitesimal generator of a contraction semigroup on \(H^2\) [2408.03182].

Invariant-subspace questions lead back to weighted composition operators. The only nonzero closed subspaces of \(H^2\) invariant under every \(W_t\), \(0<t<1\), are
\[
z^k H^2,\qquad k\ge 0.
\]
Since \(R_{\delta_t}=W_t\), it follows that the only subspaces invariant under every \(R_\mu\) are also of the form \(z^kH^2\) [2408.03182].

A recent several-variable extension replaces \(H^2(\mathbb D)\) by the Drury–Arveson space \(H^2_d(\mathbb B_d)\), with orthonormal basis
\[
e_\alpha=\sqrt{\frac{|\alpha|!}{\alpha!}}\,z^\alpha.
\]
For each coordinate \(j\), the multivariable Cesàro operator is defined on monomials by
\[
C_j z^\alpha
=
\sum_{k=0}^\infty \frac{1}{|\alpha|+k+1}\,z^{\alpha+k e_j},
\]
and the paper proves the exact norm
\[
\|C_j\|=2.
\]
The several-variable Rhaly-type operator attached to \((a_n)\) is then
\[
R_{(j,a_n)} z^\alpha=\sum_{k=0}^\infty a_k\,z^{\alpha+k e_j},
\]
with factorization
\[
R_{(j,a_n)}=D_{(j,(n+1)a_n)}\,C_j.
\]
If
\[
L=\lim_{n\to\infty}(n+1)a_n
\]
exists, then \(R_{(j,a_n)}\) is bounded if \(L<\infty\), unbounded if \(L=\infty\), and compact if \(L=0\). Moreover,
\[
\|R_{(j,a_n)}\|\le 2\sup_n |(n+1)a_n|,
\qquad
\sigma_p(R_{(j,a_n)})\subset \{a_0,a_1,\dots\},
\]
and in the compact case
\[
\sigma(R_{(j,a_n)})\subset \{0\}\cup \{a_0,a_1,\dots\}.
\]
The analysis uses the block decomposition
\[
H^2_d=\bigoplus_{\alpha:\alpha_j=0} M_\alpha,
\qquad
M_\alpha=\overline{\operatorname{span}}\{z^{\alpha+k e_j}:k\ge 0\},
\]
which reduces many questions to one-variable blocks [2606.25885].

Several open directions are explicitly identified across the recent literature. A full iff characterization of boundedness and compactness on \(H^p\) for \(p>2\) is not yet known [2511.09201]. In weighted Hardy spaces \(H^2(\omega)\), the dyadic Schatten theory is complete under mild growth control on the factorable weight sequence \(\beta_n=1/\sqrt{\omega_n}\), but genuinely non-polynomial or highly oscillatory \(\beta\) remain subtle [2606.15875]. In one and several variables alike, hyponormality and subnormality are understood in special cases—most notably for the classical Cesàro operator—but no general characterization is available for terraced sequences \(a_n\) [2408.03182], [2606.25885].

Taken together, these results place the Rhaly operator in a distinctive position among summability-type operators. Its defining terraced geometry is elementary, yet its boundedness, compactness, spectrum, and dynamics depend delicately on the ambient space and on how the generating sequence is encoded: via mean Lipschitz regularity on Hardy and Bergman spaces, weighted tail functionals on \(H^2(\omega)\), product estimates on weighted \(c_0\), duality on Köthe spaces, moment growth in integral models, and block decompositions in several complex variables.

Source: https://www.emergentmind.com/topics/rhaly-operator