A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )

2.1 Motivation from the theory of Markov chains

Recall that l 1 ( N ) is the Banach space of all absolutely summable sequences. This means that x = ( ξ i ) i ∈ N is an element of l 1 ( N ) if and only if ∑ i ∈ N ∣ ξ i ∣ < ∞ and then ∣ ∣ x ∣ ∣ l 1 ( N ) = ∑ i ∈ N ∣ ξ i ∣ . It is a separable space and any x ∈ l 1 ( N ) can be written as ∑ i ∈ N ξ i e i , where { e i } i ∈ N is the standard Schauder basis in this space, i.e., e i = ( … , 0 , 1 , 0 , … ) with 1 in the i th coordinate. For more details about l 1 ( N ) , see [7, Chapter 7].

In [4,5], the sequence of operators ( B n ) n ∈ N , defined in l 1 ( N ) , was considered, where

The sets S n 1 , S n 2 in (2.1) partition the set { 0 , 1 , 2 , … , 2 n − 1 } and are given by

The sequences ( β n ) n ∈ N and ( α n ) n ∈ N are assumed to be sequences of real positive numbers. The operators ( B n ) n ∈ N are in fact isomorphic images of the operators from [6, p. 83] and are generators of the transition semigroups associated with two-state Markov chains. These were used by Blackwell [8] to construct a Markov chain whose all states are instantaneous. To be more precise, he considered a Markov process ( X ( t ) ) t ≥ 0 defined as follows:

where X 1 ( t ) , X 2 ( t ) , X 3 ( t ) , … is an infinite sequence of mutually independent, two-state Markov chains. The set of states of every component of X ( t ) is { 0 , 1 } and the transition semigroup e t B n , associated with X n ( t ) , is thus characterized by β n and α n . An explicit formula for this semigroup exists [6, p. 83].

Let S denote the state space of X ( t ) . Then S ⊂ { 0 , 1 } ∞ , and this last set is uncountable by the known Cantor’s diagonal argument. So in general S may be uncountable. However, if we assume that the sequences ( β n ) n ∈ N , ( α n ) n ∈ N satisfy (the first condition by Blackwell)

then, with probability one, elements of S are sequences with only a finite number of 1’s. Therefore, S is countable and a bijection between S and N exists, and it is given in [5]. Moreover, the condition (2.4) also guarantees that the transition semigroup P ( t ) , associated with X ( t ) , is well defined. This means that it is a strongly continuous semigroup of contractions, and it is given by

The fact that ∏ n = 1 ∞ e t B n is the transition semigroup of X ( t ) follows from the independence of its components. The second condition introduced by Blackwell requires that

and it implies that all states of X ( t ) are instantaneous, i.e., the Q -matrix of the process has −∞ in all the diagonal entries. Incidentally, the condition (2.6), together with (2.4), implies also that ∑ n = 1 ∞ α n = + ∞ .

From these two conditions, it follows the generator of P ( t ) , denoted as A gen , is densely defined and unbounded. In [5], it was found, by the application of [3, Proposition 2.7], that A gen = A ¯ , where

Here, D ( A ) denotes the domain of A . Because A is not bounded, D ( A ) ≠ l 1 ( N ) . For example, any x with a finite number of non-zero components does not belong to D ( A ) . It should be added that in this case, the operator A ∗ defined as the limit of ∑ n = 1 N B n ∗ , when N → ∞ , is not densely defined in l ∞ ( N ) . Thus, we cannot apply the proposition from [3] and conclude the continuity of the adjoint semigroup P ∗ ( t ) .

Blackwell was interested in the case in which all states of (2.3) are instantaneous, and his example is one of the few such known, see [6, p. 82] and the references contained therein. This chain is so interesting because some explicit calculations, just described earlier, can be carried out that are related to it.

If instead of (2.6) we assume that ( β n ) n ∈ N and ( α n ) n ∈ N satisfy (in addition to (2.4))

then all states of X ( t ) are stable, which means the Q -matrix of the process has finite values in all its diagonal entries. Moreover, the operator A determined by (2.7) is then bounded, D ( A ) = l 1 ( N ) , and it generates the transition semigroup of X ( t ) [4]. This also follows from [3, Proposition 2.7], and the semigroup is defined by (2.5). Let us also emphasize that the condition (2.8) (or (2.6)) does not contradict (2.4).

The application of [3, Proposition 2.7] in the stable case also shows that (2.8) overrides (2.4). In other words, the condition (2.8) is sufficient for A to be bounded. Recall that (2.4) ensures that the set of states of X ( t ) is countable, i.e., it is a Markov chain.

Therefore, when the condition (2.8) holds true, we can omit (2.4). The operator A and the semigroup P ( t ) are still well defined and P ( t ) = e t A . However, because we no longer assume (2.4), the connection between P ( t ) and Markov chains is lost. The state space of (2.3) could be uncountable, and X ( t ) would be a Markov process. It is not clear what P ( t ) then describes. Currently, I am not able to provide any interpretation. Similarly, the meaning of B n from definition 1, as well as A in Theorem 3.3, is unclear. Nevertheless, from the point of view of semigroups, these objects are well defined.

2.2 Definition of the B n ’s

This special form of the operators given by (2.1) suggests how to generalize them so that they still commute and generate semigroups of contractive operators. The goal of this article is to prove that this generalization works and the construction is based on a finite collection of sequences of numbers, so we begin with them.

Let an integer d ≥ 2 be fixed and suppose that there are given d ⋅ ( d − 1 ) sequences of positive numbers, denoted by ( β i , j n ) n ∈ N , i.e.,

These numbers, for fixed n , can be thought of as off-diagonal entries in a d × d matrix, see Examples 1, 2. Based on (2.9), define ( β i , i n ) n ∈ N as follows:

The numbers β i , i n are diagonal entries, in the matrix analogy. It should be added that the matrix analogy is only perfect for n = 1 . From (2.10), it follows that if n is fixed, then ∑ j = 1 d β i , j n = 0 , for every i = 1 , 2 , … , d .

The sets S n 1 , S n 2 , determined by (2.2), partition the set Z 2 n − 1 = { 0 , 1 , 2 , … , 2 n − 1 } into two parts of equal size. In a similar way, we introduce a partition of Z d n − 1 = { 0 , 1 , 2 , … , d n − 1 } , denoted as S n 1 , S n 2 , … , S n d , into d parts of equal size. Namely, define

In other words, ∣ S n k ∣ = d n − 1 for k = 1 , 2 , … , d and

The sets S n 1 , S n 2 , …, S n d can also be defined recursively and in order to do this let us first introduce some notation. If S is a subset of Z , where Z stands for the set of integers, and a ∈ Z , then S + a denotes the set { s + a : s ∈ S } . So { 1 , 2 } + 3 = { 4 , 5 } . With a little abuse of notation, we also introduce

This should not lead to misunderstandings, since all indices in a vector ( ξ i ) or in a sum ∑ j = j 1 j 2 ξ j are assumed to be integers. With this notation, we write

The recursive definition of (2.11) would be to assume S n 1 = { 1 , 2 , 3 , … , d n − 1 } and

where the sum taken modulus d n ensures that 0 ∈ S n d . We can now give the following definition.

In this article, we use the convention that if in a sum ∑ j = j 1 j 2 ξ j we have j 2 < j 1 , then the sum equals zero. For example, taking i = 1 in (2.12), since S n 1 = { 1 , 2 , … , d n − 1 } , we obtain

Furthermore, it is worth noting that if i ∈ [ 1 , d n ] with i mod d n ∈ S n k , then both i − j d n − 1 ∈ [ 1 , d n ] , for j = 1 , 2 , … , k − 1 , as well as i + j d n − 1 ∈ [ 1 , d n ] , for j = 1 , 2 , … , d − k . This follows from the fact that i lies between ( k − 1 ) d n − 1 + 1 and k d n − 1 . Therefore, in the first case, we have

and, in the second case,

In a similar way, if i ∈ S l , where S l = [ 1 , d n ] + l d n , for some integer l ≥ 1 , and i mod d n ∈ S n k , then i − j d n − 1 ∈ S l , for j = 1 , 2 , … , k − 1 and i + j d n − 1 ∈ S l , for j = 1 , 2 , … , d − k .

We draw an important conclusion from these observations. Namely, the one that the (infinite) matrix of B n , denoted by M ( B n ) , can be written in the following way:

where 0 denotes the d n × d n matrix with all entries zero and M ( B ˜ n ) is the d n × d n matrix of B n truncated to the first d n coordinates. In other words, it is the matrix of the map B ˜ n : R d n → R d n defined as follows:

where η i is given by (2.12). In this case, x = ( ξ i ) i ∈ [ 1 , d n ] . The fact that M ( B n ) is the matrix of B n simply means

So it is clear that B n is a bounded linear operator on l 1 ( N ) , and using the triangle inequality, we obtain an estimate

where c n is given by

3.1 Proofs of (1.1)