Return to Article Details The eigenstructure of some positive linear operators

The eigenstructure of some positive linear operators

Antonio Attalienti $^{*}$ Ioan Raşa $^{* *}$

February 24, 2014.

$^{*}$ Department of Business and Law-University of Bari, Via Camillo Rosalba 53, 70124, Bari, Italy, e-mail: antonio.attalienti@uniba.it.

$^{* *}$ Department of Mathematics-Technical University of Cluj-Napoca str. Memorandumului 28, 400114, Cluj-Napoca, Romania, e-mail: Ioan.Rasa@math.utcluj.ro.

Of concern is the study of the eigenstructure of some classes of positive linear operators satisfying particular conditions. As a consequence, some results concerning the asymptotic behaviour as $t \to + \infty$ of particular strongly continuous semigroups $(T (t))_{t \geq 0}$ expressed in terms of iterates of the operators under consideration are obtained as well. All the analysis carried out herein turns out to be quite general and includes some applications to concrete cases of interest, related to the classical Beta, Stancu and Bernstein operators.

MSC. Primary 41A36; Secondary 47D06.

Keywords. Positive linear operators, eigenvalues and eigenpolynomials, iterates and series of positive linear operators, strongly continuous semigroups, asymptotic behaviour.

1 Introduction and notation

The present paper is devoted to the study of the eigenstructure of some classes of positive linear operators $L_{n}$ acting on the Banach lattice $C ([0, 1])$ of all real-valued continuous functions on $[0, 1]$ , endowed with the uniform norm $‖ \cdot ‖_{\infty}$ and the usual order.

In order to pursue our main results, we adopt some assumptions over the $L_{n}$ ’s. Some of them (see (3) and (4) at the beginning of Section 2) encircle our analysis in a general scheme of investigation initiated by Altomare and continued and developed originally and extensively, in different frameworks, by his school, dealing with the strong interplay between positive linear operators and strongly continuous semigroups: without attempting to be exhaustive in this respect, we confine ourselves to citing [1]-[7], [9]-[11], [14], [15] and all the references quoted therein.

Further conditions over the $L_{n}$ ’s, gathered together into three groups,
namely Case I, Case II and Case III, are needed to our purposes; as the reader will quickly realize, such additional assumptions, far from being somewhat artificial, turn out to be shared by classical positive linear operators of continuous and discrete type, namely the Beta, the Stancu and the Bernstein operators.

Indeed, the whole set of conditions provides a nice eigenstructure and, in this sense, we confirm and expand what already addressed in [ 7 , Remark 2.6 ] .

The paper is organized as follows: in Section 2 we study the eigenstructure of our operators, indicating the eigenvalues and the corresponding eigenpolynomials by quite simple techniques, which should however be compared with those employed in [ 8 ] and [ 11 ] .

The same analysis is carried out with respect to the differential operator $W$ quoted in (2.1) and to the strongly continuous semigroup $(T (t))_{t \geq 0}$ written as limit of iterates of $L_{n}$ as in (2.2).

In Section 3, proceeding along the lines illustrated in [ 7 , Section 3 ] and sketched, though inside a simpler context, in [ 6 , Theorem 2.2 ] , we focus our attention upon the asymptotic behaviour of the semigroup $(T (t))_{t \geq 0}$ , namely upon the limit $lim_{t \to + \infty} T (t) f (f \in C ([0, 1]))$ and to its possible interplay with the limits

lim_{n \to + \infty} lim_{k \to + \infty} L_{n}^{k} f, lim_{n \to + \infty} L_{n}^{k (n)} f,

1.1

$(k (n))_{n \geq 1}$ being a sequence of positive integers satisfying $k (n) / n \to + \infty$ as $n \to + \infty$ and $L_{n}^{k}$ denoting the iterate of $L_{n}$ of order $k (n, k \geq 1)$ . The limits in (1.1) are involved in an overiteration procedure, introduced and developed by the authors in [ 7 ] and which will be our leading mark in Section 3.

Occasionally, we also touch upon the convergence of the series

\sum_{k = 0}^{+ \infty} L_{n}^{k} f

for suitable functions $f$ .

As a significant application, we recapture, as particular cases, some results about the limits of the semigroups expressed in terms of iterates of the Beta, Stancu and Bernstein operators.

The notation used throughout the paper are quite standard in approximation theory and needs no particular or preparatory indication.

Therefore we shall confine ourselves to list only the most important ones: for any integer $r \geq 0$ let us set $e_{r} (x) := x^{r}, x \in [0, 1] .$

In the sequel $Π$ will denote the subalgebra of all polynomials on $[0, 1]$ : more specifically, we shall often deal with the space $Π_{r}$ of all polynomials on $[0, 1]$ of degree at most $r = 0, 1, \dots$ If $p \in Π, \deg p$ is the degree of $p$ . As usual, if $k \geq 1$ is an integer, $C^{k} ([0, 1])$ is the vector space of all real-valued $k$ -times continuously differentiable functions on $[0, 1]$ . Finally, if $x$ is a real number, then the integer part of $x$ will be denoted by $[x]$ .

Other notation which are not encompassed above, shall be specified at each occurrence.

2 Eigenvalues and eigenpolynomials

Throughout this section we shall deal with positive linear operators $L_{n} : C ([0, 1]) \to C ([0, 1]) (n \geq 1)$ acting on the Banach space $(C ([0, 1]), ‖ \cdot ‖_{\infty})$ and satisfying the following:

For each $n \geq 1$ and $r \geq 1 L_{n} e_{r}$ is a polynomial of degree $r$ with positive leading coefficient, i.e., $L_{n} e_{r} = a_{n, r} e_{r} + \dots$ , with $a_{n, r} \geq 0$ ; moreover $L_{n} e_{0} = a_{n, 0} e_{0}$ with $a_{n, 0} > 0$ .
The limit

$l_{r} := lim_{n \to + \infty} (a_{n, r})^{n}, r \geq 0,$

exists and is finite.
(Voronovskaja-type result) For any $u \in C^{2} ([0, 1])$ we have

$Label '1' multiply defined$
2.2

where explicitly $W u (x) := a (x) u^{″} (x) + b (x) u^{'} (x), a, b \in C ([0, 1]), x \in [0, 1]$ .
For every $f \in C ([0, 1])$ and $t \geq 0$ the limit

$T (t) f := lim_{n \to + \infty} L_{n}^{[n t]} f$
2.3

exists in $C ([0, 1])$ , and $(T (t))_{t \geq 0}$ is a $C_{0}$ -semigroup on $C ([0, 1])$ with infinitesimal generator $(A, D (A))$ such that $C^{2} ([0, 1]) \subset D (A)$ and $A u = W u$ for any $u \in C^{2} ([0, 1])$ .

From (1) it clearly follows that each $L_{n}$ is bounded with $‖ L_{n} ‖ = ‖ L_{n} e_{0} ‖_{\infty} = a_{n, 0}$ ; moreover each $Π_{r}$ is invariant under $L_{n}$ and consequently the same happens under $W$ and the semigroup $(T (t))_{t \geq 0}$ as well, due to (2.1), (2.2) and the closedness of $Π_{r}$ itself. In particular $W e_{1} = b \in Π_{1}$ and $W e_{2} = 2 a + 2 b e_{1}$ , whence $a \in Π_{2}$ .

As outlined in the introduction, besides these general assumptions we have to impose some further conditions over the coefficients $a_{n, j}, l_{j}$ as well as over the degree of the polynomials $L_{n} e_{r}$ , gathered together into three groups, namely Case I, Case II and Case III.

Case I

$1 = a_{n, 0} > a_{n, 1} > \dots > 0, n \geq 1$ .
$1 = l_{0} > l_{1} > \dots > 0$ .

Case II

$\deg L_{n} e_{r} = min {n, r}, n \geq 1, r \geq 0$ .
$1 = a_{n, 0} > a_{n, 1} > \dots > a_{n, r} > 0, n \geq r \geq 0$ .
$1 = l_{0} > l_{1} > \dots > 0$ .

Case III

$\deg L_{n} e_{r} = min {n, r}, n \geq 1, r \geq 0$ .
$1 = a_{n, 0} = a_{n, 1} > a_{n, 2} > \dots > a_{n, r} > 0, n \geq r \geq 0$ .
$1 = l_{0} = l_{1} > l_{2} > \dots > 0$ .

Remark 2.1

(i) All the assumptions (1)-(6) are satisfied by the classical Beta operators $B_{n}$ , introduced by Lupaş in [ 13 ] and studied for instance, as far as our investigation is concerned, in [ 4 ] , [ 5 ] and [ 7 ] : in particular, in [ 5 , Example 3.1 ] , the explicit expression of the coefficients $a_{n, r}$ and of the limits $l_{r}$ may be found.
In addition, the related differential operator $W$ defined in (3) and its interplay with a strongly continuous semigroup has been completely investigated in [ 4 , Theorem 2.10 ] .
Note that each $B_{n}$ maps $Π_{r}$ into itself for any $r \geq 0$ , hence $B_{n} (Π) \subset Π$ , even if its whole range $R (B_{n})$ is different from $Π$ .
The classical Stancu operators $S_{n}$ fulfill all the assumptions (1)-(4) and the ones listed in Case II : we refer the reader, e.g., to [ 7 , Section 4 ] , where a result about the related Voronovskaja-type formula and the existence of a strongly continuous semigroup expressed in terms of iterates of the $S_{n}$ ’s has been stated. For the reader’s convenience we recall that explicitly, for fixed $α \geq 1 / 2$ and $β \geq α + 1 / 2$ , the $n$ -th Stancu operator $S_{n}$ is given by

S_{n} f := \sum_{i = 0}^{n} b_{n i} f (\frac{i + α}{n + β})

for all $f \in C ([0, 1])$ , where $b_{n i} (x) := (\binom{n}{i}) x^{i} (1 - x)^{n - i}, x \in [0, 1]$ .

Accordingly, it is not a difficult task to show that $a_{n, 0} = 1$ and

a_{n, r} = \frac{n (n - 1) \times \dots \times (n - (r - 1))}{(n + β)^{r}}, n \geq r \geq 1, l_{r} = e^{- \frac{r (r + 2 β - 1)}{2}}, r \geq 0.

Finally, we remark how the last Case III is of particular interest since all the conditions (10)-(12) enclosed herein, together with (1)-(4), hold true for the Bernstein operators $B_{n}$ . In this particular situation one easily computes $a_{n, 0} = 1$ and

a_{n, r} = \frac{n (n - 1) \times \dots \times (n - (r - 1))}{n^{r}}, n \geq r \geq 1, l_{r} = e^{- \frac{r (r - 1)}{2}}, r \geq 0.

For a rather complete analysis about the related Voronovskaja formula and the existence of a strongly continuous semigroup expressed in terms of the iterates of the $B_{n}$ ’s see, for instance, [ 1 ] - [ 3 ] , [ 9 ] , [ 10 ] , [ 14 ] and [ 15 ] .

(ii) Under the assumptions quoted in Cases II and III, each $L_{n}$ maps continuous functions into polynomials in $Π_{n}$ ; indeed, choose $f \in C ([0, 1])$ and a sequence $(p_{r})_{r \geq 1}$ in $Π$ such that $lim_{r \to + \infty} p_{r} = f$ . For a fixed $n \geq 1$ we get

L_{n} f = L_{n} (lim_{r \to + \infty} p_{r}) = lim_{r \to + \infty} L_{n} p_{r},

which gives $L_{n} f \in Π_{n}$ since $Π_{n}$ is closed and $L_{n} p_{r} \in Π_{n}$ for $r$ large enough by virtue of (7) or (10).

(iii) In Case III the additional information $L_{n} e_{1} = e_{1}$ is available, too; indeed, for any given $n \geq 1$ , by assumption (11) we already know that $L_{n} e_{1} = e_{1} + α_{n} e_{0}$ for some $α_{n} \in R$ . On the other hand $0 \leq e_{1} \leq e_{0}$ and therefore $0 \leq e_{1} + α_{n} e_{0} \leq e_{0}$ because obviously $L_{n} e_{0} = e_{0}$ . Evaluating the last inequality in $x = 0$ and $x = 1$ gives $α_{n} = 0$ , as desired.
In turn, according to (4), the condition $L_{n} e_{1} = e_{1}$ gives $T (t) e_{1} = e_{1}$ for all $t \geq 0$ whence $W e_{1} \equiv 0$ , i.e., $b \equiv 0$ (see the discussion straight after (2.2)). (iv) Incidentally observe that in any case $L_{n} e_{0} = e_{0}$ and therefore each $L_{n}$ has norm $1$ .â–¡

Now we may establish our first result.

Theorem 2.2

Let $n \geq 1$ and $r \geq 0$ be fixed; in Cases II and III we further assume $n \geq r$ . Then $L_{n} : Π_{r} \to Π_{r}$ has $r + 1$ eigenvalues

a_{n, 0}, a_{n, 1}, \dots, a_{n, r} .

2.4

To each $a_{n, j}$ there corresponds a monic eigenpolynomial $p_{n, j}$ with $\deg p_{n, j} = j$ , i.e., $L_{n} p_{n, j} = a_{n, j} p_{n, j}, j = 0, \dots, r .$

Proof â–¼

The assumptions over $n$ and $r$ guarantee that in any case $L_{n} (Π_{r}) \subset Π_{r}$ so that on account of (1), (7) and (10) we may rightly write $L_{n} : Π_{r} \to Π_{r}$ ; with respect to the basis ${e_{0}, \dots, e_{r}}$ the matrix of $L_{n}$ is upper triangular and is given by

M_{n, r} = (\begin{array}{ccc} a_{n, 0} & \hdots & \hdots \\ 0 & a_{n, 1} & \hdots \\ ⋮ & ⋱ & ⋮ \\ 0 & \hdots & a_{n, r} \end{array}) .

2.5

Clearly the corresponding eigenvalues are those indicated in (2.3). Moreover, they are distinct in Cases I and II on account of (5) and (8), and an easy computation allows to determine uniquely the related eigenpolynomials $p_{n, j}, j = 0, \dots, r$ . In Case III the eigenpolynomials corresponding to the eigenvalues $a_{n, 0} = a_{n, 1} = 1$ are given respectively by $p_{n, 0} = e_{0}$ and $p_{n, 1} = e_{1}$ by virtue of (iii) and (iv) in Remark 2.1, whereas the remaining ones $p_{n, j}, j = 2, \dots, r$ may be found in the usual way.

The following theorem is devoted to the analysis of the eigenstructure of the differential operator $W$ .

Theorem 2.3

The differential operator $W : Π \to Π$ defined in (2.1) has eigenvalues $\log l_{0}, \log l_{1}, \dots$ and corresponding monic eigenpolynomials $p_{0}, p_{1}, \dots$ with $\deg p_{j} = j,$ i.e., $W p_{j} = (\log l_{j}) p_{j}, j = 0, 1, \dots$ In addition, for any $j$ we have

lim_{n \to + \infty} p_{n, j} = p_{j} uniformly on [0, 1],

2.6

$p_{n, j}$ being defined in the previous theorem.

Proof â–¼

Since $Π_{0} \subset Π_{1} \subset \dots \subset Π$ , it is enough to show that, for an arbitrary but fixed $r \geq 0$ , the operator $W : Π_{r} \to Π_{r}$ has eigenvalues $\log l_{0}, \log l_{1}, \dots, \log l_{r}$ with monic eigenpolynomials $p_{0}, p_{1}, \dots, p_{r} (\deg p_{j} = j)$ satisfying (2.5) for any $j = 0, 1, \dots, r .$

To this purpose, let us fix, once and for all, an integer $r \geq 0$ and denote by $U_{r}$ the matrix of the operator $W : Π_{r} \to Π_{r}$ with respect to the basis ${e_{0}, \dots, e_{r}}$ . Writing down (2.1) for $e_{0}, \dots, e_{r}$ and denoting by $I$ the unit matrix, we deduce a matricial version of the Voronovskaja formula, namely

lim_{n \to + \infty} n (M_{n, r} - I) = U_{r} coordinatewise,

2.7

which soon implies $lim_{n \to + \infty} M_{n, r} = I$ , i.e.,

lim_{n \to + \infty} a_{n, j} = 1, j = 0, 1, \dots, r

2.8

on account of (2.4). But then, recalling (2), we get

l_{j} = lim_{n \to + \infty} (1 + (a_{n, j} - 1))^{n} = e^{lim_{n \to + \infty} n (a_{n, j} - 1)},

and consequently

lim_{n \to + \infty} n (a_{n, j} - 1) = \log l_{j}, j = 0, 1, \dots, r,

2.9

which, together with (2.4) and (2.6), allows to conclude that the matrix $U_{r}$ is upper triangular and is given by

U_{r} = (\begin{array}{ccc} \log l_{0} & \hdots & \hdots \\ 0 & \log l_{1} & \hdots \\ ⋮ & ⋱ & ⋮ \\ 0 & \hdots & \log l_{r} \end{array}) .

2.10

Thus $U_{r}$ (or equivalently $W$ ) has $\log l_{0}, \log l_{1}, \dots, \log l_{r}$ as its eigenvalues: they are distinct in Cases I and II and to each of them there corresponds a unique eigenpolynomial $p_{j}, j = 0, 1, \dots, r .$ In Case III we find $\log l_{0} = \log l_{1} = 0$ with corresponding eigenpolynomials $p_{0} = e_{0}$ and $p_{1} = e_{1}$ and a standard computation allows to determine uniquely the remaining ones $p_{j}, j = 2, \dots, r .$

In order to prove (2.5), for the above fixed $r$ let $n \geq r$ . By Theorem 2.2 we already know that

n (L_{n} - I) p_{n, j} = n (a_{n, j} - 1) p_{n, j}, j = 0, 1, \dots, r,

2.11

which may be rephrased by saying that each $p_{n, j}$ is an eigenpolynomial of $n (L_{n} - I)$ corresponding to the eigenvalue $n (a_{n, j} - 1)$ .

For each $j = 0, 1, \dots, r$ let $p_{n, j} = e_{j} + x_{j - 1}^{n} e_{j - 1} + \dots + x_{0}^{n} e_{0}$ . According to (2.10) the unknowns $x_{j - 1}^{n}, \dots, x_{0}^{n}$ may be uniquely determined by solving the $(r + 1) \times (r + 1)$ system

(n (M_{n, r} - I) - n (a_{n, j} - 1) I) \cdot (x_{0}^{n}, \dots, x_{j - 1}^{n}, 1, 0, \dots, 0)^{T} = (0, \dots, 0)^{T},

2.12

which obviously reduces to a $(j + 1) \times (j + 1)$ system

\begin{aligned} (\begin{array}{ccccc} n (a_{n, 0} - a_{n, j}) & \hdots & \hdots & \hdots & \hdots \\ 0 & n (a_{n, 1} - a_{n, j}) & \hdots & \hdots & \hdots \\ 0 & 0 & ⋱ & \hdots & \hdots \\ ⋮ & ⋮ & \hdots & n (a_{n, j - 1} - a_{n, j}) & ⋮ \\ 0 & 0 & \hdots & 0 & 0 \end{array}) \cdot (\begin{array}{c} x_{0}^{n} \\ ⋮ \\ ⋮ \\ x_{j - 1}^{n} \\ 1 \end{array}) = (\begin{array}{c} 0 \\ ⋮ \\ ⋮ \\ 0 \\ 0 \end{array}) . \end{aligned}

On the other hand, if we consider an eigenpolynomial $p_{j} = e_{j} + y_{j - 1} + \dots + y_{0} e_{0}$ of $U_{r}$ , then the unknowns $y_{j - 1}, \dots, y_{0}$ are the solution of the $(r + 1) \times (r + 1)$ system

(U_{r} - (\log l_{j}) I) \cdot (y_{0}, \dots, y_{j - 1}, 1, 0, \dots, 0)^{T} = (0, \dots, 0)^{T},

2.14

which obviously reduces to a $(j + 1) \times (j + 1)$ system

\begin{aligned} (\begin{array}{ccccc} \log l_{0} - \log l_{j} & \hdots & \hdots & \hdots & \hdots \\ 0 & \log l_{1} - \log l_{j} & \hdots & \hdots & \hdots \\ 0 & 0 & ⋱ & \hdots & \hdots \\ ⋮ & ⋮ & \hdots & \log l_{j - 1} - \log l_{j} & ⋮ \\ 0 & 0 & \hdots & 0 & 0 \end{array}) \cdot (\begin{array}{c} y_{0} \\ ⋮ \\ ⋮ \\ y_{j - 1} \\ 1 \end{array}) = (\begin{array}{c} 0 \\ ⋮ \\ ⋮ \\ 0 \\ 0 \end{array}) . \end{aligned}

From (2.6) and (2.8) we infer that the matrix of the coefficients of the system (2.11) tends coordinatewise as $n \to + \infty$ to the analogous matrix of the system (2.13) and therefore the same happens for the matrices of the coefficients in the systems (2.12) and (2.14), respectively. It immediately follows that $lim_{n \to + \infty} x_{i}^{n} = y_{i}$ for any $i = 0, \dots, j - 1$ and consequently

lim_{n \to + \infty} p_{n, j} = p_{j}, uniformly on [0, 1], j = 0, 1, \dots, r,

2.16

which concludes the proof.

The next corollary deals with the eigenstructure of the strongly continuous semigroup $(T (t))_{t \geq 0}$ quoted in (2.2).

Corollary 2.4

For any $t \geq 0 T (t) : Π \to Π$ has eigenvalues $l_{0}^{t}, l_{1}^{t}, \dots$ with the same eigenpolynomials $p_{0}, p_{1}, \dots$ from Theorem 2.3.

Proof â–¼

Simply observe that, on account of Theorem 2.2, for any $n \geq 1, j \geq 0$ and $t \geq 0$ one gets $L_{n}^{[n t]} p_{n, j} = a_{n, j}^{[n t]} p_{n, j}$ ; passing to the limit as $n \to + \infty$ yields

T (t) p_{j} = l_{j}^{t} p_{j}

2.17

by virtue of (2), (2.2), (2.5) and the boundedness of each $L_{n}$ .

It seems useful to display the situation described so far about eigenpolynomials in the following tables, where we adopt the same notation used in Theorems 2.2 and 2.3.

Case I

\begin{array}{cccccc} p_{1, 0} & p_{1, 1} & p_{1, 2} & p_{1, 3} & \hdots \\ p_{2, 0} & p_{2, 1} & p_{2, 2} & p_{2, 3} & \hdots \\ p_{3, 0} & p_{3, 1} & p_{3, 2} & p_{3, 3} & \hdots \\ p_{4, 0} & p_{4, 1} & p_{4, 2} & p_{4, 3} & \hdots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ ↓ & ↓ & ↓ & ↓ & ↓ & (n \to + \infty) \\ p_{0} & p_{1} & p_{2} & p_{3} & \hdots \end{array}

2.18

Cases II and III

\begin{array}{ccccccc} p_{1, 0} & p_{1, 1} \\ p_{2, 0} & p_{2, 1} & p_{2, 2} \\ p_{3, 0} & p_{3, 1} & p_{3, 2} & p_{3, 3} \\ p_{4, 0} & p_{4, 1} & p_{4, 2} & p_{4, 3} & p_{4, 4} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ ↓ & ↓ & ↓ & ↓ & ↓ & ↓ & (n \to + \infty) \\ p_{0} & p_{1} & p_{2} & p_{3} & p_{4} & \hdots \end{array}

2.19

3 Asymptotic behaviour of the semigroup $(T (\lowercase t))_{\lowercase t \geq 0}$
and overiteration

Let us open this section with the following two general results, which shall be useful in the sequel, covering, perhaps, an interest on their own.

Proposition 3.1

Let $T$ be an arbitrary bounded positive linear operator on a Banach space $(X, ‖ \cdot ‖)$ and suppose that, for a given $x \in X$ , there exists $P x := lim_{k \to + \infty} T^{k} x \in X$ . Then we have:

$P^{2} x = P x .$
There exists $P T x \in X$ and $T P x = P x = P T x .$

Proof â–¼

If $k \geq 1$ , then

T^{k} P x = T^{k} (lim_{j \to + \infty} T^{j} x) = lim_{j \to + \infty} T^{k + j} x = P x,

3.20

which, for $k = 1$ and $k \to + \infty$ , gives $T P x = P x$ and $P^{2} x = P x$ , respectively.

Lastly, $P x = lim_{k \to + \infty} T^{k + 1} x = lim_{k \to + \infty} T^{k} (T x)$ , i.e., $P T x$ exists in $X$ and is equal to $P x$ . The proof is now complete.

Proposition 3.2

Under the same assumptions and notation of Proposition 3.1, the following are equivalent:

The series $\sum_{k = 0}^{+ \infty} T^{k} x$ is convergent in $X$ .
There exists $y \in X$ with $P y = 0$ such that $x = y - T y$ .
There exists $z \in X$ such that $P z$ exists in $X$ and $x = z - T z$ .

Proof â–¼

$(a) ⟹ (b)$ : Let us set $y = \sum_{k = 0}^{+ \infty} T^{k} x \in X$ ; then for all $j \geq 1$ one has

T^{j} y = \sum_{k = 0}^{+ \infty} T^{k + j} x = \sum_{i = 0}^{+ \infty} T^{i} x - \sum_{i = 0}^{j - 1} T^{i} x = y - \sum_{i = 0}^{j - 1} T^{i} x,

and therefore $lim_{j \to + \infty} T^{j} y = y - y = 0$ , i.e., $P y = 0$ . Moreover, for any $k \geq 1$

(I - T) (I + T + \dots + T^{k}) x = (I - T^{k + 1}) x,

3.21

and letting $k \to + \infty$ immediately gives $y - T y = x$ , since clearly
$lim_{k \to + \infty} T^{k + 1} x = 0$ .

Since $(b) ⟹ (c)$ is obvious, let us pass to show that $(c) ⟹ (a)$ . To this aim, replacing $x$ in (3.2) with the $z$ given in $(c)$ soon yields

(I + T + \dots + T^{k}) x = z - T^{k + 1} z \underset{k \to + \infty}{⟶} z - P z \in X,

3.22

so that the series $\sum_{k = 0}^{+ \infty} T^{k} x$ is convergent, as desired.

Remark 3.3

As a deeper insight, let us consider, as in $(c)$ , $x = z - T z$ such that $P z$ exists in $X$ . If we put $y := z - P z$ , then one readily gets $P y = P z - P^{2} z = 0$ and $y - T y = x$ as a direct application of Proposition 3.1.â–¡

Now let us pass to the main objective of this section, i.e., the study of the asymptotic behaviour of the semigroup $(T (t))_{t \geq 0}$ by means of the overiteration procedure involving limits in (1.1). To attain our main goals, we have to assume that henceforth each $L_{n}$ has a totally positive kernel in the sense of Karlin, as described, in great details, in [ 12 ] .

As pointed out in [ 5 ] , we are dealing with a quite natural hypothesis, by no means breaking the generality of our investigation, since commonly fulfilled in concrete cases by most of the classical positive linear operators occurring in approximation theory and, however, intimately connected to the issue about the preservation of higher order convexity and Lipschitz classes: for a rather complete analysis in this direction, we refer the reader to [ 12 ] and [ 5 ] .

Throughout the remaining of this section the discussion will be split up into two parts, the first concerning the Case I and the latter the Cases II and III.

Case I

As a preparatory material, let us start by choosing $p \in Π$ ; if $\deg p = r, r \geq 0$ , then surely by Proposition 2.2 for every fixed $n \geq 1$ the polynomial $p$ may be expanded as

p = c_{n, 0} (p) p_{n, 0} + c_{n, 1} (p) p_{n, 1} + \dots + c_{n, r} (p) p_{n, r},

3.23

where the coefficients $c_{n, j} (p), j = 0, 1, \dots, r,$ are uniquely determined and $c_{n, 0} (p)$ does not depend on $r$ . Then, for each $k \geq 1$ , we easily compute

L_{n}^{k} p = c_{n, 0} (p) e_{0} + c_{n, 1} (p) a_{n, 1}^{k} p_{n, 1} + \dots + c_{n, r} (p) a_{n, r}^{k} p_{n, r},

3.24

because $L_{n} p_{n, j} = a_{n, j} p_{n, j}, j = 1, \dots, r$ due to Theorem 2.2, $a_{n, 0} = 1$ and $p_{n, 0} = e_{0}$ .

Note that $c_{n, 0} : Π \to R$ is a linear functional; moreover, if $p \geq 0$ , then $L_{n}^{k} p \geq 0$ and letting $k \to + \infty$ in (3.5) yields $c_{n, 0} (p) \geq 0$ since $a_{n, j} < 1$ for every $j = 1, \dots r$ by assumption.

On the other hand, taking $p = e_{0}$ in (3.5) gives $c_{n, 0} (e_{0}) = 1$ .

Summing up, we have just shown that $c_{n, 0} : Π \to R$ is a positive linear functional on $(Π, ‖ \cdot ‖_{\infty})$ with $‖ c_{n, 0} ‖ = 1$ .

The application of the classical Hahn-Banach Theorem allows to extend $c_{n, 0}$ to a norm-one functional on the whole space $C ([0, 1])$ ; due to the density of $Π$ , such extension, still denoted by $c_{n, 0}$ , is unique and positive, as well.

Now let us set $P_{n} : C ([0, 1]) \to C ([0, 1])$ as

P_{n} f = c_{n, 0} (f) e_{0} for every f \in C ([0, 1]) .

3.25

Of course $P_{n}$ is a norm-one positive linear operator on $C ([0, 1])$ such that $lim_{k \to + \infty} L_{n}^{k} p = P_{n} p$ for all $p \in Π$ and this, by a density argument, leads soon to

lim_{k \to + \infty} L_{n}^{k} f = P_{n} f for every f \in C ([0, 1]) .

3.26

Now we are in a position to state the following result.

Theorem 3.4

Under the above-mentioned assumptions and notation, there exists $lim_{t \to + \infty} T (t) f := S f$ for every $f \in C ([0, 1])$ ; moreover

lim_{n \to + \infty} P_{n} f = S f, lim_{n \to + \infty} L_{n}^{k (n)} f = S f

3.27

for every $f \in C ([0, 1])$ and for every sequence of positive integers $(k (n))_{n \geq 1}$ satisfying $k (n) / n \to + \infty$ as $n \to + \infty$ .

Proof â–¼

The existence of the limit of the semigroup $(T (t))_{t \geq 0}$ as $t \to + \infty$ follows from Theorem 2.2 in [ 6 ] , indicating, in addition, that $S f$ is a constant function. For the remainder of the proof simply apply Corollary 2.3 and Theorem 2.2 in [ 7 ] .

Remark 3.5

(i) If, in particular, the $L_{n}$ ’s are the Beta operators $B_{n}$ (see (i) in Remark (2.1)), then $S$ is explicitly described in [ 7 , Theorem 3.1 ] . [(ii)] If for a given $n \geq 1 c_{n, 0} (p) = 0$ in (3.5), then

\sum_{k = 0}^{+ \infty} L_{n}^{k} p = \frac{c_{n, 1} (p)}{1 - a_{n, 1}} p_{n, 1} + \dots + \frac{c_{n, r} (p)}{1 - a_{n, r}} p_{n, r} = (I - L_{n})^{- 1} p,

where $(I - L_{n})^{- 1} : Π \to Π .$

(iii) By using (3.5) an estimate of the speed of convergence in (3.7) for every fixed $n \geq 1$ as far as the polynomials are concerned may be obtained; indeed, since $1 = a_{n, 0} > a_{n, 1} > \dots > 0, n \geq 1$ , one has

‖ L_{n}^{k} p - P_{n} p ‖ \leq (| c_{n, 1} (p) | \cdot ‖ p_{n, 1} ‖ + \dots + | c_{n, r} (p) | \cdot ‖ p_{n, r} ‖) a_{n, 1}^{k}

for all $k \geq 1$ and $p \in Π$ .â–¡

Actually, something more can be said still in the framework of Case I, as stated in the two next propositions.

Proposition 3.6

For a fixed $n \geq 1$ and $p \in Π$ we have

Label '1' multiply defined

3.28

$R (I - L_{n})$ denoting the range of $I - L_{n}$ .

Proof â–¼

According to (3.5) and the subsequent discussion, if $\deg p = r, r \geq 0$ , then $lim_{k \to + \infty} L_{n}^{k} p = 0$ if and only if $c_{n, 0} (p) = 0$ which, on account of (3.4), implies

p = c_{n, 1} (p) p_{n, 1} + \dots + c_{n, r} (p) p_{n, r},

i.e., $p = (I - L_{n}) z$ where, by definition,

z := \frac{c_{n, 1} (p)}{1 - a_{n, 1}} p_{n, 1} + \dots + \frac{c_{n, r} (p)}{1 - a_{n, r}} p_{n, r} .

3.29

Proposition 3.7

For a fixed $n \geq 1$ and $p \in Π$ we have

\sum_{k = 0}^{+ \infty} L_{n}^{k} p is convergent ⟺ p \in R ((I - L_{n})^{2}) .

3.30

Proof â–¼

By virtue of Proposition 3.2, the series $\sum_{k = 0}^{+ \infty} L_{n}^{k} p$ is convergent if and
only if $p = y - L_{n} y$ for some $y \in Π$ such that $lim_{k \to + \infty} L_{n}^{k} y = 0,$ which, by virtue of (3.9), equals to $y = s - L_{n} s$ for some $s \in R (I - L_{n})$ . But then $p \in R ((I - L_{n})^{2})$ and the proof is now fully performed.

Now we have to consider the Cases II and III which will be treated simultaneously.

Cases II and III

Also in these cases a result word-for-word identical to Theorem 3.4 (with $P_{n} f$ defined below in (3.15) or (3.16)) may be achieved. Therefore we do not restate the assertion, passing soon to show the proof which runs similar, in fact simpler.

Indeed, due to [ 6 , Theorem 2.2 ] , in both cases there exists

lim_{t \to + \infty} T (t) f := S f for every f \in C ([0, 1]),

3.31

$S f$ being a constant function in Case II (when $l_{1} < 1, l_{2} < 1$ ), a polynomial in $Π_{1}$ in Case III (when $l_{2} < 1$ and $b \equiv 0$ ; see (iii) in Remark 2.1).

Now consider $f \in C ([0, 1])$ and $n \geq 1$ ; on account of (ii) in Remark 2.1 we already know that $L_{n} \in Π_{n}$ and therefore an expansion analogous to (3.4)

L_{n} f = c_{n, 0} (f) p_{n, 0} + c_{n, 1} (f) p_{n, 1} + \dots + c_{n, n} (f) p_{n, n}

3.32

is just available; consequently, for any $k \geq 2$ , keeping in mind Theorem 2.2 one has

L_{n}^{k} f = c_{n, 0} (f) a_{n, 0}^{k - 1} p_{n, 0} + c_{n, 1} (f) a_{n, 1}^{k - 1} p_{n, 1} + \dots + c_{n, n} (f) a_{n, n}^{k - 1} p_{n, n}

3.33

so that we may easily investigate the limit as $k \to + \infty$ . More precisely, in Case II we have

P_{n} f := lim_{k \to + \infty} L_{n}^{k} f = c_{n, 0} (f) e_{0}

3.34

since $a_{n, 0} = 1$ and $p_{n, 0} = e_{0}$ , whereas in Case III

P_{n} f := lim_{k \to + \infty} L_{n}^{k} f = c_{n, 0} (f) e_{0} + c_{n, 1} (f) e_{1}

3.35

since now $a_{n, 0} = a_{n, 1} = 1$ and $p_{n, 0} = e_{0}, p_{n, 1} = e_{1}$ .

Note that the limit $S f$ of the semigroup in (3.12) is invariant under each $L_{n}$ in both cases (simply recall (iii) in Remark 2.1); therefore, exactly as in Case I, we are in a position to apply Corollary 2.3 and Theorem 2.2 in [ 7 ] and hence

lim_{n \to + \infty} P_{n} f = S f, lim_{n \to + \infty} L_{n}^{k (n)} f = S f

3.36

hold true for every $f \in C ([0, 1])$ and for every sequence of positive integers $(k (n))_{n \geq 1}$ satisfying $k (n) / n \to + \infty$ as $n \to + \infty$ .

Remark 3.8

We also point out that, if $f \in C ([0, 1])$ and $n \geq 1$ , then from (3.14) it easily follows that

\sum_{k = 0}^{+ \infty} L_{n}^{k} f is convergent ⟺ P_{n} f = 0 (= lim_{k \to + \infty} L_{n}^{k} f),

where $P_{n} f$ is defined in (3.15) or (3.16), accordingly.â–¡

Remark 3.9

As already mentioned, the Stancu operators $S_{n}$ fall within Case II; an explicit expression of $S f$ in (3.12) and (3.17) in this particular case may be found in [ 7 , Formula (4.1) and Theorem 4.1 ] .

An application of [ 7 , Theorem 2.2 ] supplying a relationship analogous to (3.17) for the Bernstein operators is indicated in [ 15 , Theorem 2.3 ] . However in this context, which corresponds to Case III, something more can be said in general about $S f$ . Indeed, we already know that $S : C ([0, 1]) \to Π_{1}$ is a positive linear projection.

Furthermore, $T (t) e_{0} = e_{0}$ and $T (t) e_{1} = e_{1}$ for all $t \geq 0$ (see (2.2) and (iii) in Remark 2.1) so that $S e_{0} = e_{0}$ and $S e_{1} = e_{1}$ by (3.12).

For any given $x \in [0, 1]$ let us set $η_{x} (f) := S f (x)$ for any $f \in C ([0, 1])$ . Then $η_{x}$ is a probability Radon measure with barycenter $x$ ; in particular, $η_{0} = δ_{0}$ and $η_{1} = δ_{1}$ , where $δ_{0}$ and $δ_{1}$ denote the Dirac measures at $0$ and $1$ , respectively.

Now, for an arbitrary $f \in C ([0, 1])$ , we may write down $S f = α e_{0} + β e_{1}$ for suitable reals $α$ and $β$ which entails $S f (0) = α$ and $S f (1) = α + β$ ; on the other hand, $S f (0) = δ_{0} (f) = f (0)$ and $S f (1) = δ_{1} (f) = f (1)$ . In conclusion, we get $α = f (0), β = f (1) - f (0)$ so that the projection $S$ is uniquely determined by

S f (x) = (1 - x) f (0) + x f (1) (f \in C [0, 1]), x \in [0, 1]),

recapturing, in this way, a well-known result, scattered in the literature, about the limit, as $t \to + \infty$ , of the semigroup $(T (t))_{t \geq 0}$ expressed through iterates of the classical Bernstein operators: we refer, for instance, to [ 3 , Remark 3.11.1 ] and [ 14 , Theorem 3.10 ] .â–¡

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