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A research problem on which I am currently working requires a construction in topological dynamics of the following type:

Let $T \colon X \to X$ be a continuous transformation of a compact metric space which contains at least two points, and let $(a_n)$ be an absolutely summable real sequence which is not the zero sequence. When can we guarantee that there exists a continuous function $f \colon X \to \mathbb{C}$ such that $\sum_{n=1}^\infty a_n f \circ T^n$ is not a constant function?

The above question would seem to amount to a question about the spectral behaviour of the composition operator $U_T \colon C(X) \to C(X)$ defined by $U_Tf(x):=f(T(x))$. It is easy to show that the eigenvalues of $U_T$ form a subgroup of the unit circle. If $U_T$ has an eigenvalue which is not a root of unity then the eigenvalues are dense in the unit circle, so given a fixed sequence $(a_n)$ an easy Fourier analysis argument allows us to choose an eigenfunction $f$ for which $\sum_{n=1}^\infty a_n f \circ T^n$ is not constant. On the other hand, in some cases where $U_T$ has a root of unity as an eigenvalue, there are sequences such that the series converges to a constant for all $f$: for example, if $X$ contains just two points then the sequence given by $a_1=a_2=1$ and $a_n=0$ for $n \geq 3$ has this property. The case of most interest, then, is that in which $U_T$ has no eigenvalues except $1$, 1$and no eigenfunctions other than the constant function, which is referred to as topological weak mixing. Specifically, I ask: Is topological weak mixing of$T$sufficient to guarantee the existence of$f$in the first question? This suggests to me the following more general functional-analytic question, which (by considering the action of$U_T$on the quotient of$C(X)$modulo the subspace of constant functions) would be sufficient for a positive answer to the above: Let$L$be a bounded linear operator acting on an infinite-dimensional Banach space$B$, with the spectrum of$L$being$\{1\}$and the norm of$L$being$1$. Does When does there necessarily exist$x \in B$such that the sequence$\{L^nx \colon n \geq 0\}$is$\omega$-linearly independent, i.e. for all nonzero absolutely summable sequences$(a_n)$, the sum$\sum_{n=1}^\infty a_n L^nx$is nonzero? Thanks in advance! 3 added 30 characters in body A research problem on which I am currently working requires a construction in topological dynamics of the following type: Let$T \colon X \to X$be a continuous transformation of a compact metric space which contains at least two points, and let$(a_n)$be an absolutely summable real sequence which is not the zero sequence. When can we guarantee that there exists a continuous function$f \colon X \to \mathbb{C}$such that$\sum_{n=1}^\infty a_n f \circ T^n$is not a constant function? The above question would seem to amount to a question about the spectral behaviour of the composition operator$U_T \colon C(X) \to C(X)$defined by$U_Tf(x):=f(T(x))$. It is easy to show that the eigenvalues of$U_T$form a subgroup of the unit circle. If$U_T$has an eigenvalue which is not a root of unity then the eigenvalues are dense in the unit circle, so given a fixed sequence$(a_n)$an easy Fourier analysis argument allows us to choose an eigenfunction$f$for which$\sum_{n=1}^\infty a_n f \circ T^n$is not constant. On the other hand, in some cases where$U_T$has a root of unity as an eigenvalue, there are sequences such that the series converges to a constant for all$f$: for example, if$X$contains just two points then the sequence given by$a_1=a_2=1$and$a_n=0$for$n \geq 3$has this property. The case of most interest, then, is that in which$U_T$has no eigenvalues except$1$, which is referred to as topological weak mixing. Specifically, I ask: Is topological weak mixing of$T$sufficient to guarantee the existence of$f$in the first question? This suggests to me the following more general functional-analytic question, which (by considering the action of$U_T$on the quotient of$C(X)$modulo the subspace of constant functions) would be sufficient for a positive answer to the above: Let$L$be a bounded linear operator acting on an infinite-dimensional Banach space$B$, with the spectrum of$L$being$\{1\}$. \{1\}$ and the norm of $L$ being $1$. Does there necessarily exist $x \in B$ such that the sequence $\{L^nx \colon n \geq 0\}$ is $\omega$-linearly independent, i.e. for all nonzero absolutely summable sequences $(a_n)$, the sum $\sum_{n=1}^\infty a_n L^nx$ is nonzero?

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A research problem on which I am currently working requires a construction in topological dynamics of the following type:

Let $T \colon X \to X$ be a continuous transformation of a compact metric space which contains at least two points, and let $(a_n)$ be an absolutely summable real sequence which is not the zero sequence. When can we guarantee that there exists a continuous function $f \colon X \to \mathbb{C}$ such that $\sum_{n=1}^\infty a_n f \circ T^n$ is not a constant function?

The above question would seem to amount to a question about the spectral behaviour of the composition operator $U_T \colon C(X) \to C(X)$ defined by $U_Tf(x):=f(T(x))$. It is easy to show that the eigenvalues of $U_T$ form a subgroup of the unit circle. If $U_T$ has an eigenvalue which is not a root of unity then the eigenvalues are dense in the unit circle, so given a fixed sequence $(a_n)$ an easy Fourier analysis argument allows us to choose an eigenfunction $f$ for which $\sum_{n=1}^\infty a_n f \circ T^n$ is not constant. On the other hand, in some cases where $U_T$ has a root of unity as an eigenvalue, there are sequences such that the series converges to a constant for all $f$: for example, if $X$ contains just two points then the sequence given by $a_1=a_2=1$ and $a_n=0$ for $n \geq 3$ has this property. The case of most interest, then, is that in which $U_T$ has no eigenvalues except $1$, which is referred to as topological weak mixing. Specifically, I ask:

Is topological weak mixing of $T$ sufficient to guarantee the existence of $f$ in the first question?

This suggests to me the following more general functional-analytic question, which (by considering the action of $U_T$ on the quotient of $C(X)$ modulo the subspace of constant functions) would be sufficient for a positive answer to the above:

Let $L$ be a bounded linear operator acting on a an infinite-dimensional Banach space $B$, with the spectrum of $L$ being $\{1\}$. Does there necessarily exist $x \in B$ such that the sequence $\{L^nx \colon n \geq 0\}$ is $\omega$-linearly independent, i.e. for all nonzero absolutely summable sequences $(a_n)$, the sum $\sum_{n=1}^\infty a_n L^nx$ is nonzero?