Divergence Behavior of Sequences of Linear Operators with Applications

Abstract

In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach–Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., $\ell ^p$, $1\le p < \infty$, C[0, 1], $L^p$, $1< p <\infty$, as well as Paley–Wiener and Bernstein spaces, have the s-property. Finally, consequences for the convergence behavior of sampling series and system approximation processes are shown.

Appendix: Equivalence of Questions 1 and 2

In this section, we show the equivalence of Questions 1 and 2. We start with reducing Question 1 to a simpler question.

Theorem 11

The answer to Question 1 is positive for arbitrary separable Banach spaces $B_1, B_2$ if and only if the answer to Question 1 is positive for arbitrary closed subspaces $B_1, B_2$ of C[0, 1].

Proof

$\Rightarrow$: This direction is trivial, because every closed subspace of C[0, 1] is a Banach space.

$\Leftarrow$: Let $B_1$ and $B_2$ two arbitrary separable Banach spaces, and assume that the assumptions of Question 1 are fulfilled. According to the Banach–Mazur theorem [4, 24] there exist closed subspaces $H_1, H_2$ of C[0, 1] and isometric isomorphisms $U_1, U_2$ such that $U_l(B_l) = H_l$, $l=1,2$.

Let $T^c = U_2 \circ T \circ U_1^{-1}$ and $T_N^c = U_2 \circ T_N \circ U_1^{-1}$, $N \in \mathbb {N}$, both of which map from $H_1$ into $H_2$. We have $||T^c ||_{H_1 \rightarrow H_2} = ||T ||_{B_1 \rightarrow B_2}$ and $||T_N^c ||_{H_1 \rightarrow H_2} = ||T_N ||_{B_1 \rightarrow B_2}$, $N \in \mathbb {N}$. According to our assumption there exists a dense subset $\mathcal {M}\subset B_1$ such that

$$\begin{aligned} \lim _{N \rightarrow \infty } ||T_N f - T f ||_{B_2} = 0 \end{aligned}$$

for all $f \in \mathcal {M}$. It follows that $\mathcal {M}_c \mathrel {\mathop {:}}=U_1(\mathcal {M})$ is dense in $H_1$ and that for all $f \in \mathcal {M}_c$ we have

$$\begin{aligned} \lim _{N \rightarrow \infty } ||T_N^c f - T^c f ||_{H_2} = 0 . \end{aligned}$$

The assertion for arbitrary closed subspaces of C[0, 1], which we assume to be true, gives that there exists an infinite dimensional closed subspace $\underline{H_1} \subset H_1$ such that

$$\begin{aligned} \limsup _{N \rightarrow \infty } ||T_N^c f ||_{H_2} = \infty \end{aligned}$$

for all $f \in \underline{H_1}\setminus \{0\}$. It follows that $\underline{B_1} = U_1^{-1} \underline{H_1}$ is an infinite dimensional closed subspace of $B_1$, and that we have

$$\begin{aligned} \limsup _{N \rightarrow \infty } ||T_N f ||_{B_2} = \infty \end{aligned}$$

for all $f \in \underline{B_1} \setminus \{0\}$. $\square$

We want to further reduce the question by showing that the structure of the space $B_2$ is not particularly significant; it suffices to consider $B_2 = \mathbb {C}$. This leads us to the following question.

Question 3

Let $B_1$ be a closed subspace of C[0, 1] and $\{\psi _N\}_{N \in \mathbb {N}}$ a sequence of continuous linear functionals on $B_1$, satisfying

• (A1’) $\limsup _{N \rightarrow \infty } ||\psi _N ||_{B_1 \rightarrow \mathbb {C}} = \infty$, and

• (A2’) there exists a continuous linear functional $\psi :B_1 \rightarrow \mathbb {C}$ as well as a dense subset $\mathcal {M}\subset B_1$ such that $\lim _{N \rightarrow \infty } \psi _N f = \psi f$ for all $f \in \mathcal {M}$.

Is the set

$$\begin{aligned} \left\{ f \in B_1 :\limsup _{N \rightarrow \infty } |\psi _N f |= \infty \right\} \end{aligned}$$

spaceable?

Theorem 12

The answer to Question 1 is positive if and only if the answer to Question 3 is positive.

Proof

$\Rightarrow$: If the answer to Question 1 is positive, then the assertion is true for arbitrary separable Banach spaces $B_1$ and $B_2$, and thus, in particular for $B_1$ being an arbitrary closed subspace of C[0, 1] and $B_2 = \mathbb {C}$.

$\Leftarrow$: Let $B_1, B_2$ be two arbitrary closed subspaces of C[0, 1]. The set of all $f \in B_1$ satisfying $\limsup _{N \rightarrow \infty } ||T_N f ||_{C[0,1]} = \infty$ is a residual set. Hence, there exist a $f \in B_1$ and two sequences $\{N_k\}_{k \in \mathbb {N}} \subset \mathbb {N}$ and $\{t_k\}_{k \in \mathbb {N}} \subset [0,1]$, such that

$$\begin{aligned} \lim _{k \rightarrow \infty } |(T_{N_k} f)(t_k) |= \infty . \end{aligned}$$

Further, there exists a $t_* \in [0,1]$ and a subsequence $\{k_l\}_{l \in \mathbb {N}}$ such that

$$\begin{aligned} \lim _{l \rightarrow \infty } |t_* - t_{k_l} |= 0 . \end{aligned}$$

We consider the functionals

$$\begin{aligned} \psi _l f = (T_{N_{k_l}} f)(t_{k_l}) , \quad l \in \mathbb {N}. \end{aligned}$$

For $f \in \mathcal {M}$ we have that for all $\epsilon >0$ there exists a $l_0 = l_0(\epsilon )$ such that

$$\begin{aligned} ||T_{N_{k_l}} f - Tf ||_{C[0,1]} < \epsilon \end{aligned}$$

for all $l \ge l_0$. Since Tf is continuous, there exists a $l_1=l_1(\epsilon )$ such that

$$\begin{aligned} |(Tf)(t_*) - (Tf)(t_{k_l}) |< \epsilon \end{aligned}$$

for all $l \ge l_1$. Hence, for all $l \ge \max \{l_0,l_1\}$ we have

$$\begin{aligned} |(T_{N_{k_l}} f)(t_{k_l}) - (Tf)(t_*) |< 2\epsilon . \end{aligned}$$

Therefore, we have for all $f \in \mathcal {M}$ that

$$\begin{aligned} \lim _{l \rightarrow \infty } \psi _l f = \psi f , \end{aligned}$$

where $\psi f = (Tf)(t_*)$. Thus, the set

$$\begin{aligned} \left\{ f \in B_1:\limsup _{l \rightarrow \infty } |\psi _l f |= \infty \right\} \end{aligned}$$

is spaceable. Further, we have

$$\begin{aligned} ||T_{N_{k_l}} f ||_{C[0,1]} \ge |\psi _l f |, \quad l \in \mathbb {N}. \end{aligned}$$

and therefore spaceability of the set

$$\begin{aligned} \left\{ f \in B_1 :\limsup _{N \rightarrow \infty } ||T_N f ||_{B_2} = \infty \right\} . \end{aligned}$$

$\square$

Next we show that it is sufficient to consider specific sequences of linear functionals, which is the the final simplification.

Corollary 8

The answer to Question 1 is positive if and only if Question 3 can be answered positively for all sequences of functionals $\{\psi _N\}_{N \in \mathbb {N}}$ with $\lim _{N \rightarrow \infty } \psi _N(f) = 0$ for all $f \in \mathcal {M}$.

Proof

Choose $\psi _{N}^* = \psi _N- \psi$. $\square$