Rhaly Operator: Terraced Summability in Analysis

• Rhaly operator is a lower‐triangular terraced operator defined via a scalar sequence that multiplies initial partial sums, generalizing the classical Cesàro operator.
• It bridges summability theory and analysis by connecting function spaces such as Hardy, Bergman, and Köthe through explicit factorization and boundedness criteria.
• Its spectral, compactness, and dynamical properties depend sensitively on the weight sequence, inspiring ongoing research in operator theory.

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 (Galanopoulos et al., 12 Nov 2025, Bellavita et al., 27 Mar 2025, Doğan, 27 Jul 2025, Bellavita et al., 14 Jun 2026, Pilla, 24 Jun 2026).

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 $(\eta_n)$0-indexed convention used on weighted null sequence spaces and Köthe spaces,

$(\eta_n)$1

and

$(\eta_n)$2

The two conventions are notational variants of the same terraced construction (Bellavita et al., 27 Mar 2025, Rani et al., 12 Oct 2025, Doğan, 27 Jul 2025).

For analytic functions, the coefficient action is central. If

$(\eta_n)$3

then

$(\eta_n)$4

where $(\eta_n)$5 denotes Hadamard convolution,

$(\eta_n)$6

This identity is one of the defining analytic features of the operator (Galanopoulos et al., 12 Nov 2025).

Several equivalent operator factorizations recur throughout the literature. On $(\eta_n)$7,

$(\eta_n)$8

where $(\eta_n)$9 is the discrete Hardy operator $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$0 and $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$1 is diagonal, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$2 (Bellavita et al., 27 Mar 2025). In Cesàro-based formulations,

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$3

with $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$4 the classical Cesàro matrix and $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$5 (Gallardo-Gutiérrez et al., 2024). On weighted Hardy spaces $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$6, the operator becomes a factorable matrix $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$7 in the orthonormal basis $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$8, with

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$9

connecting Rhaly theory directly to the theory of factorable matrices (Bellavita et al., 14 Jun 2026).

A persistent misconception is to identify the Rhaly operator with a Toeplitz, Nörlund, or Cauchy-product convolution operator. The coefficient rule

$f(z)=\sum_{k\ge 0} a_k z^k$0

is not of the form $f(z)=\sum_{k\ge 0} a_k z^k$1, 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(z)=\sum_{k\ge 0} a_k z^k$2; boundedness of the multiplier $f(z)=\sum_{k\ge 0} a_k z^k$3 for $f(z)=\sum_{k\ge 0} a_k z^k$4 does not imply boundedness of $f(z)=\sum_{k\ge 0} a_k z^k$5 (Galanopoulos et al., 12 Nov 2025).

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(z)=\sum_{k\ge 0} a_k z^k$6 as the decisive criterion for boundedness and compactness on classical analytic function spaces (Galanopoulos et al., 12 Nov 2025). For $f(z)=\sum_{k\ge 0} a_k z^k$7 and $f(z)=\sum_{k\ge 0} a_k z^k$8, the mean Lipschitz space $f(z)=\sum_{k\ge 0} a_k z^k$9 is characterized by

$a_n=\eta_n=1/(n+1)$0

and its little-oh variant $a_n=\eta_n=1/(n+1)$1 requires

$a_n=\eta_n=1/(n+1)$2

On Hardy spaces $a_n=\eta_n=1/(n+1)$3, the main dichotomy is at $a_n=\eta_n=1/(n+1)$4. If $a_n=\eta_n=1/(n+1)$5, then

$a_n=\eta_n=1/(n+1)$6

and

$a_n=\eta_n=1/(n+1)$7

If $a_n=\eta_n=1/(n+1)$8, boundedness implies $a_n=\eta_n=1/(n+1)$9, and compactness implies $n$0, but sufficiency is presently established only under the stronger hypotheses

$n$1

for some $n$2 (Galanopoulos et al., 12 Nov 2025). The same work states explicitly that a full if-and-only-if characterization for $n$3 remains open.

For unweighted Bergman spaces $n$4, the same paper obtains exact criteria for all $n$5: $n$6 More generally, on weighted Bergman spaces $n$7 with $n$8, mean Lipschitz conditions suffice for boundedness and compactness, and in the range $n$9 they are also necessary (Galanopoulos et al., 12 Nov 2025).

The $0$0-based Hardy theory is sharper in Schatten-class terms. For the one-variable Rhaly matrix $0$1 on $0$2, which is identified with $0$3, boundedness is equivalent to each of the following: $0$4

$0$5

and

$0$6

Compactness is equivalent to the corresponding vanishing dyadic condition, or equivalently to $0$7. For $0$8,

$0$9

equivalently $\ell^2$0. In particular,

$\ell^2$1

(Bellavita et al., 27 Mar 2025).

On weighted Hardy spaces $\ell^2$2, boundedness and compactness are governed by factorable-matrix parameters. Writing

$\ell^2$3

and

$\ell^2$4

one has

$\ell^2$5

with

$\ell^2$6

Compactness is characterized by the corresponding little-oh conditions (Bellavita et al., 14 Jun 2026).

This weighted theory recovers familiar spaces. For $\ell^2$7, $\ell^2$8, boundedness is equivalent to

$\ell^2$9

which is equivalent to

$$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$0

and therefore to $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$1. For $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$2, $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$3, boundedness is again equivalent to the same tail condition. For the critical Dirichlet space $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$4, the criterion changes to

$$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$5

and for Schatten class $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$6 an additional logarithmic factor appears: $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$7 By contrast, on $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$8, $$R_{n,k}=\alpha_n \quad (0\le k\le n),\qquad R_{n,k}=0\quad (k>n),$$9, and $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$0 with $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$1, the dyadic condition

$(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$2

characterizes $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$3-membership for $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$4, independently of $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$5 and of $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$6 (Bellavita et al., 14 Jun 2026).

The classical Cesàro operator sits exactly at the threshold between boundedness and compactness in many of these settings. For $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$7, the operator is bounded on $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$8 and $(R_\alpha x)(k)=\alpha_k\sum_{j=0}^k x(j).$9, but on $(\eta_n)$00 it is neither compact nor Schatten-class. The paper on Hardy and Bergman spaces further isolates the borderline $(\eta_n)$01 decay: if $(\eta_n)$02 and $(\eta_n)$03, then $(\eta_n)$04 is bounded on $(\eta_n)$05; however, there exists a sequence with $(\eta_n)$06 for which $(\eta_n)$07 is not bounded on $(\eta_n)$08 when $(\eta_n)$09. For decreasing nonnegative $(\eta_n)$10, boundedness on $(\eta_n)$11, $(\eta_n)$12, is equivalent to $(\eta_n)$13 (Galanopoulos et al., 12 Nov 2025).

3. Weighted null sequence spaces and fine spectral structure

A second major line of work studies Rhaly operators on weighted null sequence spaces

$(\eta_n)$14

where $(\eta_n)$15 is strictly positive. If $(\eta_n)$16, then $(\eta_n)$17 with equivalent norms; the genuinely weighted regime is $(\eta_n)$18 (Rani et al., 12 Oct 2025, Patra et al., 2023).

For bounded, strictly positive weights $(\eta_n)$19, the exact boundedness criterion for

$(\eta_n)$20

is

$(\eta_n)$21

and then

$(\eta_n)$22

Compactness is characterized by

$(\eta_n)$23

In the equal-weight case $(\eta_n)$24, this is often written using

$(\eta_n)$25

with $(\eta_n)$26 and $(\eta_n)$27 (Patra et al., 2023, Rani et al., 12 Oct 2025).

Several sufficient conditions are weight-driven rather than sequence-driven. If $(\eta_n)$28 is strictly positive and

$(\eta_n)$29

then bounded $(\eta_n)$30 implies boundedness of $(\eta_n)$31, and $(\eta_n)$32 implies compactness. Geometric weights $(\eta_n)$33, $(\eta_n)$34, fall directly into this regime. The literature also emphasizes that this decay condition is sufficient, not necessary: for example, with

$(\eta_n)$35

where $(\eta_n)$36 and $(\eta_n)$37, one still has

$(\eta_n)$38

hence compactness, even though $(\eta_n)$39 and the simple sufficient condition does not apply (Rani et al., 12 Oct 2025).

The compact case admits a complete spectral description under standard monotonicity and positivity hypotheses. Assume $(\eta_n)$40 is a sequence of non-zero real numbers, $(\eta_n)$41 is a bounded decreasing sequence of strictly positive reals, and $(\eta_n)$42. Writing

$(\eta_n)$43

the spectrum satisfies

$(\eta_n)$44

$(\eta_n)$45

Moreover,

$(\eta_n)$46

and the spectral radius is

$(\eta_n)$47

Goldberg’s classification places all nonzero spectral points in class $(\eta_n)$48 and $(\eta_n)$49 in class $(\eta_n)$50 (Rani et al., 12 Oct 2025).

A related but distinct weighted-$(\eta_n)$51 spectral theory arises when the asymptotic parameter

$(\eta_n)$52

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

$(\eta_n)$53

then

$(\eta_n)$54

while

$(\eta_n)$55

and, for $(\eta_n)$56,

$(\eta_n)$57

In this regime the spectrum outside $(\eta_n)$58 is contained in the closed disk centered at $(\eta_n)$59 with radius $(\eta_n)$60, yielding the bound $(\eta_n)$61 (Patra et al., 2023).

These weighted-sequence-space results show that terraced lower-triangular structure can lead either to fully discrete compact spectra $(\eta_n)$62, 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 $(\eta_n)$63 is a Köthe matrix, the Köthe space is

$(\eta_n)$64

with dual

$(\eta_n)$65

For a terraced sequence $(\eta_n)$66, the associated operator satisfies

$(\eta_n)$67

A general continuity criterion states that

$(\eta_n)$68

is well defined and continuous if and only if $(\eta_n)$69 and, for every $(\eta_n)$70, there exists $(\eta_n)$71 such that

$(\eta_n)$72

If $(\eta_n)$73 and $(\eta_n)$74 for some $(\eta_n)$75, continuity follows; if moreover $(\eta_n)$76 is Montel, then $(\eta_n)$77 is compact (Doğan, 27 Jul 2025).

The paper "The Rhaly Operators on Köthe Spaces" specializes this theory to power series spaces. For an increasing sequence $(\eta_n)$78,

$(\eta_n)$79

and

$(\eta_n)$80

The standard identifications

$(\eta_n)$81

place Rhaly operators into holomorphic function theory (Doğan, 27 Jul 2025).

On infinite-type spaces, the behavior is especially rigid. If $(\eta_n)$82, then

$(\eta_n)$83

is continuous and compact; likewise, if $(\eta_n)$84, then

$(\eta_n)$85

is continuous and compact. On finite-type spaces, continuity from $(\eta_n)$86 to $(\eta_n)$87 follows under the comparison condition

$(\eta_n)$88

and for a nuclear $(\eta_n)$89 one has the sharp self-map criterion

$(\eta_n)$90

In particular, the classical Cesàro operator is not compact on $(\eta_n)$91 because the sequence $(\eta_n)$92 does not belong to the strong dual $(\eta_n)$93 (Doğan, 27 Jul 2025).

The same paper develops topological and ergodic properties. If $(\eta_n)$94 for some $(\eta_n)$95, then $(\eta_n)$96 is $(\eta_n)$97-topologizable on $(\eta_n)$98. On $(\eta_n)$99, every $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$00 with $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$01 is $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$02-topologizable and power bounded. On nuclear $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$03, power boundedness is characterized by

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$04

Using the ABR and KT criteria cited in that paper, one obtains the equivalence, on nuclear $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$05, between power boundedness, mean ergodicity, uniform mean ergodicity, and Cesàro boundedness together with

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$06

pointwise (Doğan, 27 Jul 2025).

Integral representations tie this locally convex theory back to the terraced matrix. For $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$07,

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$08

and the associated terrace sequence is given by the Taylor coefficients of $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$09. An analogous formula holds on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$10. The paper deduces from this that on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$11, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$12 is compact for every $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$13, while on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$14 compactness is equivalent to $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$15 belonging to the strong dual of $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$16 (Doğan, 27 Jul 2025).

5. Generalized Cesàro operators and related Rhaly families

A closely related family is the generalized Cesàro, or Rhaly, operator

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$17

with matrix entries

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$18

At $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$19 this reduces to the classical Cesàro operator. For $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$20, the subdiagonals are geometrically tilted (Curbera et al., 2023).

On a broad class of translation-invariant solid Banach lattices with natural basis, including all separable rearrangement-invariant sequence spaces, various weighted $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$21 and $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$22 spaces, Cesàro spaces $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$23, dual-type spaces $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$24, and Bachelis spaces $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$25, one has a uniform compactness phenomenon: $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$26 where $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$27 is the diagonal operator $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$28, and $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$29 is compact while $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$30 is bounded. Hence $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$31 is compact for every $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$32. Its spectrum is fully explicit: $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$33

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$34

whenever the canonical basis is a basis. In the non-separable Cesàro space $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$35, by contrast, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$36 lies in the residual spectrum (Curbera et al., 2023).

The explicit eigenvectors are given, for each $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$37, by a sequence $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$38 with $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$39 for $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$40, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$41, and

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$42

so the ratio tends to $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$43 and the vector belongs to the decreasing-majorant space $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$44, hence to all spaces under consideration (Curbera et al., 2023).

A weighted-$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$45 analogue yields the same spectral pattern in the compact case. For compact generalized Cesàro operators

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$46

on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$47, boundedness and compactness are characterized by the sequence

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$48

with boundedness equivalent to $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$49 and compactness equivalent to $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$50. In the compact case,

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$51

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$52

with the same Goldberg classification pattern as for compact Rhaly operators (Rani et al., 12 Oct 2025).

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 $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$53, yielding a family that is compact for $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$54 in many spaces where the classical Cesàro operator $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$55 is not compact.

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

When the terrace sequence is a moment sequence

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$56

the Rhaly operator admits an integral representation on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$57: $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$58 Equivalently,

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$59

This formulation supports spectral and invariant-subspace results unavailable for a general terrace sequence (Gallardo-Gutiérrez et al., 2024).

For bounded $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$60 on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$61, letting

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$62

the point spectrum is characterized by

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$63

Consequently,

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$64

This recovers the classical Cesàro fact $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$65 for Lebesgue measure, since then $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$66 and $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$67; it also yields $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$68 for a point mass $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$69, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$70 (Gallardo-Gutiérrez et al., 2024).

The adjoint can be substantially larger spectrally. If $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$71, then $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$72. Moreover, if $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$73 satisfies

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$74

then the open disk

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$75

For measures with density $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$76, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$77, this reproduces the Cesàro-type disk

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$78

The same paper also proves a numerical-range localization: $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$79 and deduces that $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$80 is the infinitesimal generator of a contraction semigroup on $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$81 (Gallardo-Gutiérrez et al., 2024).

Invariant-subspace questions lead back to weighted composition operators. The only nonzero closed subspaces of $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$82 invariant under every $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$83, $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$84, are

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$85

Since $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$86, it follows that the only subspaces invariant under every $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$87 are also of the form $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$88 (Gallardo-Gutiérrez et al., 2024).

A recent several-variable extension replaces $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$89 by the Drury–Arveson space $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$90, with orthonormal basis

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$91

For each coordinate $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$92, the multivariable Cesàro operator is defined on monomials by

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$93

and the paper proves the exact norm

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$94

The several-variable Rhaly-type operator attached to $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$95 is then

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$96

with factorization

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$97

If

$$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$98

exists, then $$(R_a x)_n=a_n\sum_{k=0}^n x_k \qquad\text{or}\qquad (\mathcal R_{(\eta)}f)(z)=\sum_{n=0}^\infty\Big(\eta_n\sum_{k=0}^n a_k\Big)z^n$$99 is bounded if $f(z)=\sum_{k\ge 0} a_k z^k$00, unbounded if $f(z)=\sum_{k\ge 0} a_k z^k$01, and compact if $f(z)=\sum_{k\ge 0} a_k z^k$02. Moreover,

$f(z)=\sum_{k\ge 0} a_k z^k$03

and in the compact case

$f(z)=\sum_{k\ge 0} a_k z^k$04

The analysis uses the block decomposition

$f(z)=\sum_{k\ge 0} a_k z^k$05

which reduces many questions to one-variable blocks (Pilla, 24 Jun 2026).

Several open directions are explicitly identified across the recent literature. A full iff characterization of boundedness and compactness on $f(z)=\sum_{k\ge 0} a_k z^k$06 for $f(z)=\sum_{k\ge 0} a_k z^k$07 is not yet known (Galanopoulos et al., 12 Nov 2025). In weighted Hardy spaces $f(z)=\sum_{k\ge 0} a_k z^k$08, the dyadic Schatten theory is complete under mild growth control on the factorable weight sequence $f(z)=\sum_{k\ge 0} a_k z^k$09, but genuinely non-polynomial or highly oscillatory $f(z)=\sum_{k\ge 0} a_k z^k$10 remain subtle (Bellavita et al., 14 Jun 2026). 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 $f(z)=\sum_{k\ge 0} a_k z^k$11 (Gallardo-Gutiérrez et al., 2024, Pilla, 24 Jun 2026).

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 $f(z)=\sum_{k\ge 0} a_k z^k$12, product estimates on weighted $f(z)=\sum_{k\ge 0} a_k z^k$13, duality on Köthe spaces, moment growth in integral models, and block decompositions in several complex variables.