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ABSTRACT
We characterize all real matrix semigroups, indexed by the non-negative reals, which satisfy a mild boundedness assumption, without assuming continuity. Besides the continuous solutions of the semigroup functional equation, we give a description of solutions arising from non-measurable solutions of Cauchy’s functional equation. To do so, we discuss the primary decomposition and the Jordan—Chevalley decomposition of a matrix semigroup. Our motivation stems from a characterization of all multi-dimensional self-similar Gaussian Markov processes, which is given in a companion paper.
1. Introduction
It is a classical fact that all continuous matrix-valued functions satisfying the semigroup property
where is the identity matrix, are given by the maps
, where
is arbitrary. See, e.g. Problem 2.1 in (Engel & Nagel, Citation2000) and the subsequent discussion. To the best of our knowledge, few authors have considered non-continuous solutions of (Equation1.1
(1.1)
(1.1) ) in the multidimensional case d > 1. Kuczma and Zajtz (Kuczma & Zajtz, Citation1966) determine all matrix semigroups
where
is measurable. Zajtz (Zajtz, Citation1971) characterizes the matrix semigroups
indexed by rational numbers. The one-dimensional case, which is equivalent to Cauchy’s functional equation,
is well studied, on the other hand (see (Aczél, Citation1966; Bingham et al., Citation1987)). Our motivation to investigate non-continuous matrix semigroups stems from probability theory: In a companion paper, we develop a characterization of all self-similar Gaussian Markov processes. By self-similarity, the bivariate covariance function of such a multi-dimensional stochastic process can be transformed to a matrix function of a single argument, which must satisfy (Equation1.1(1.1)
(1.1) ). See Section 2 for some more details. Our main result (Theorem 2.6) determines all solutions of (Equation1.1
(1.1)
(1.1) ) which satisfy a mild boundedness assumption. In Section 3, we prepare the proof by providing a decomposition of the vector space into invariant subspaces, and establish some useful properties of the decomposition. The main proofs (in Section 5) are preceded by a discussion of the Jordan—Chevalley decomposition of a semigroup, in Section 4. In Appendix A, we give two auxiliary results on Cauchy’s functional equation.
2. Preliminaries and main result
To motivate our investigation, we first recall some facts about Gaussian stochastic processes (see, e.g. Lifshits, Citation2012; Nourdin, Citation2012, for more information). Let be a d-dimensional real centered Gaussian process. The covariance of X is a matrix-valued function
satisfying
and uniquely characterizes the law of the process. Suppose that X is self-similar, and that is the identity matrix. In terms of the covariance function, self-similarity means that
for
and some self-similarity parameter H > 0. By a classical criterion (Doob, Citation1953) [Theorem V.8.1], X is a Markov process if and only if its covariance function satisfies
Upon introducing , self-similarity allows to reduce (Equation2.1
(2.1)
(2.1) ) to
This observation has been used in dimension d = 1 to prove that certain Gaussian processes do not have the Markov property (see, e.g. (Nourdin, Citation2012), [Theorem 2.3]). Unifying and generalizing these isolated results in a companion paper, we obtain a classification of all d-dimensional self-similar Gaussian Markov processes. A full classification requires finding all solutions of (Equation1.1(1.1)
(1.1) ), without assuming continuity. The boundedness assumption 2.5 stated below causes no problems, though. To state our main result, define the rotation matrix
and, for even k and a function , the block-diagonal matrix
consisting of rotation matrices. In our statements, ν will denote some non-measurable (equivalently, non-continuous) solution of (Equation1.3
(1.3)
(1.3) ). The following assumption is in force throughout the paper.
Assumption 2.1.
In the following, V denotes a real d-dimensional vector space equipped with an inner product .
Definition 2.2.
We write L(V) for the set of linear maps from V to itself. The operator norm induced by the inner product is denoted by
. If the basis is clear from the context, we will identify elements of L(V) and d × d matrices.
Definition 2.3.
We say that is a semigroup if
and
for all . Semigroups acting on
, the complexification of V, are defined by the same property.
Definition 2.4.
We say that a semigroup in L(V) is elementary if there exists an orthonormal basis such that
for some non-continuous ν satisfying Cauchy’s EquationEquation (1.3)(1.3)
(1.3) and some matrix
.
Assumption 2.5.
The semigroup satisfies
for
, where the function
is locally bounded, right-continuous at 0 and satisfies
.
We can now state our main theorem, which will be proven at the end of Section 5.
Theorem 2.6.
Let be a semigroup satisfying Assumption 2.5. Then, there exists an orthogonal decomposition
such that each Vi is invariant under g(x), and either g(x) is elementary on Vi or
for x > 0.
We call degenerate if
for x > 0.
Example 2.7.
Let , and let
be a non-continuous solution of Cauchy’s functional EquationEquation (1.3)
(1.3)
(1.3) . Then, the semigroup given by
is an example of a semigroup covered by Theorem 2.6 (by putting d = 2 and in Definition 2.4), but not the previous results mentioned at the beginning of the introduction. It illustrates that, in contrast to dimension one, a two-dimensional locally bounded semigroup need not be continuous.
3. Primary decomposition for semigroups
In this section, we discuss a decomposition of V into subspaces which are invariant for the given semigroup.
Definition 3.1.
Let S be a linear operator on the vector space V. A subspace is called S-invariant if S maps U into U.
The primary decomposition theorem from linear algebra [O’Meara et al., Citation2011, Theorem 1.5.1] decomposes a vector space into invariant subspaces for a given operator. On each subspace, the operator has a single real eigenvalue or a pair of conjugate complex eigenvalues. Instead of a single operator, we need the following version for semigroups.
Theorem 3.2.
Primary decomposition For any semigroup of linear maps acting on V, there exists a decomposition
with
such that each Vi is g(x)-invariant for all
, and for all i one of the following holds:
(1) | For all | ||||
(2) | For all |
Proof.
It is known that the primary decomposition extends to commuting sets of matrices. Indeed, the decomposition into invariant subspaces follows from Theorem 5 on p. 40 in (Jacobson, Citation1962), applied to the span of the semigroup
. Thus, we only need to argue why (1) or (2) follows from the semigroup property. Assume that each g(x) has only one eigenvalue
on some Vi. Since the g(x) commute, they share a common eigenvector vi, and thus we have
for
. Suppose
for some
. Then, we have
, and hence
, contradicting
for all
. Hence
for all
.
Theorem 3.3.
Consider a semigroup acting on
, the complexification of V. Then, there exists a decomposition
with
such that, for each i and all
, the space Vi is g(x)-invariant, and g(x) has only one eigenvalue
on Vi.
Proof.
This is an immediate consequence of Theorem 5 on p. 40 in (Jacobson, Citation1962) (cf. the preceding proof).
Definition 3.4.
We call the decomposition from Theorem 3.2 simultaneous real primary decomposition (SRPD) of V, omitting “w.r.t. g” if the semigroup is clear from the context. The component Vi is of first type in case (1), and of second type in case (2). Similarly, the simultaneous primary decomposition (SPD) is the decomposition from Theorem 3.3.
Lemma 3.5.
Let g be a semigroup acting on V, and let be a subspace of first type from the SRPD of V. Then, there exists a common eigenvector
such that
for all x > 0.
Proof.
We present an algorithm which yields the subspace of common eigenvectors. If itself is this subspace, then we are done. Otherwise there exists
such that the eigenspace
of
is a strict subspace of Vi. For any y > 0 and any
we have
It follows that is g(x)-invariant for all x > 0. Now either
consists of common eigenvectors, or we can again find
such that the eigenspace
of
in
is a strict subspace of
. Repeating this argument yields a sequence of nontrivial subspaces
whose dimensions are strictly decreasing, hence it has to terminate. Clearly, the final vector space in this sequence is the space of common eigenvectors of the semigroup.
Lemma 3.6.
Let g be a semigroup acting on V, and let Vi be a subspace from the SRPD of V such that there exists with
. Then, we have
for all x > 0. Furthermore
for all x > 0.
Proof.
By the previous lemma, there exists a common eigenvector . (Since commuting matrices are simultaneously triangularizable it follows that they share a common eigenvector
.) We have
Hence λ satisfies . Since
we have
for all
since
. Let y > 0, then there exists
such that
. We obtain
Since for all y > 0 it follows that the characteristic polynomial of g(y) satisfies
and hence, by the Cayley—Hamilton theorem,
. For
we obtain
Corollary 3.7.
Let g be a semigroup acting on V, with SRPD . For any x > 0 we have
, where
Proof.
If , then
for all
and hence Vi is of type 1. By the previous lemma, we have
If the inclusion was strict, then there would exist Vj with such that
. But since g(x) is invertible on Vj this gives a contradiction, hence we have equality.
Corollary 3.8.
Let g be a semigroup acting on V. Then, there exists a decomposition , such that
is invertible for all
and
for x > 0.
4. Multiplicative Jordan–Chevalley decomposition of semigroups
Due to Corollary 3.8, from now on we assume in most of our statements that g(x) is invertible for all . A standard result from linear algebra, the Jordan—Chevalley decomposition, asserts that any matrix A can be uniquely decomposed as
, where D is diagonalizable, N is nilpotent and D and N commute. If A is invertible, then we can express it as
with T unipotent and commuting with D.
Definition 4.1.
For an invertible linear map A on with
, the multiplicative decomposition A = DT into commuting factors with D diagonalizable and T unipotent, is called the multiplicative Jordan—Chevalley decomposition.
For background on the (multiplicative) Jordan—Chevalley decomposition, we refer to Section 15.1 in (Humphreys, Citation1975). We now analyze the structure of the multiplicative Jordan—Chevalley decomposition of a semigroup.
Theorem 4.2.
Let be a semigroup of invertible linear maps acting on
with
and let
be the multiplicative Jordan—Chevalley decomposition of each g(x). Then
and
each form a semigroup, and the two families commute with each other, i.e.
for all
.
Proof.
We can w.l.o.g. assume that , since by uniqueness of the Jordan—Chevalley decomposition
and
if
for
. Take the SPD from Theorem 3.3,
, so that each g(x) has only one eigenvalue on Vi. Denote by
the multiplicative Jordan—Chevalley decomposition g(x) restricted to Vi. Denote by
the eigenvalue of g(x) on Vi. Clearly,
. Since the
are a commuting family of matrices, they share a common eigenvector
. We have
and hence is a semigroup. Since each
is a multiple of the identity, it commutes with every linear map and hence
which shows that is also a semigroup. The result then follows, since by uniqueness
is the multiplicative Jordan—Chevalley decomposition of g(x).
Theorem 4.3.
Let be a semigroup of invertible linear maps acting on V and let
be its multiplicative Jordan—Chevalley decomposition. Then, there exist commuting real diagonalizable linear maps J(x) and commuting real nilpotent linear maps N(x) satisfying
(1) | |||||
(2) | |||||
(3) |
|
such that
and
Proof.
Let be the SRPD and let
. Assume first that Vi is of first type and
is the single positive eigenvalue of
. Then, the multiplicative Jordan—Chevalley decomposition on Vi is
. Define
. Since
is nilpotent, we have
Notice that since the logarithm converges for all unipotent matrices, the exponential map between the Lie algebra of nilpotent matrices and the Lie group of unipotent matrices is bijective (see p. 35 in (Goodman & Wallach, Citation1998)). Since is a semigroup we have
Rewrite this as
Since is nilpotent and exp is bijective, we have
For any set
, which is invertible for
. It is clear that
Applying the same idea again yields
Again by uniqueness we obtain
or equivalently
By commutativity we obtain
and hence, again by uniqueness, we have . It follows that
and
have the desired properties.
In the second case Vi is of second type. By Theorem 3.3, and since is real, Vi decomposes over
as
, where
has only one eigenvalue on each Uj and U1 is isomorphic to U2 with isomorphism given by
. By taking the principal branch of the logarithm, in the same manner as in the real case, we obtain commuting
on U1 and
on U2 satisfying Cauchy’s equation such that
Notice that , where
and
satisfies Cauchy’s functional equation on
. By Lemma A.1, we can lift any solution on
to a solution on
such that linearity is preserved. From now on denote this lift by
. Hence, if we choose any basis of U1 and its complex conjugate on U2 we obtain that
is similar to
Taking the similarity transform with the matrix
where , we obtain
Since matrix similarity over is equivalent to matrix similarity over
for two real matrices, there exists a real matrix Bi on Vi such that
Setting
and
we see that Ji and Ni satisfy the desired conditions, with and
by uniqueness of the Jordan—Chevalley decomposition. The direct sums
and
give the required matrices. Furthermore, in the case where, there exists x > 0 for which
has a complex eigenvalue
we have
where . Recall the matrix defined in (Equation2.2
(2.2)
(2.2) ). By changing the order of the basis, we have that Ui is similar to the block diagonal matrix (recall the notation (Equation2.2
(2.2)
(2.2) ))
Hence
where is the composition of Bi with some permutation matrix P.
5. Proof of the main result
After providing some final preparatory results, this section ends with the proof of Theorem 2.6. Consider again the SRPD from Theorem 3.2. Now on each Vi, g(x) has either one positive eigenvalue or two complex conjugate eigenvalues
and
, where it is possible that
for some values of x. If Vi is of first type, set
. Each νi is a solution to Cauchy’s functional EquationEquation (1.3)
(1.3)
(1.3) . Consider then the set
and partition it into equivalent solutions, according to Definition A.2. This then gives a partition of the index set
in the following manner: If
or
, then
are in the same subset of the partition. This is well-defined, since if
then
. Set
.
Definition 5.1.
We call the decomposition the partitioned SRPD of V.
Furthermore associate with each Wj one solution ηj of Cauchy’s equation such that with
. If ηj is linear we always take
. Notice that for i ≠ j we have
, hence there can be at most one Wi with
, and furthermore this is the only Wi which can have odd dimension since it contains all Vj of type 1 (recall Definition 3.4).
Theorem 5.2.
Let be a semigroup of invertible linear maps acting on V, and denote by
its partitioned SRPD. Denote by ηi the solution associated with Wi. If ηi is non-continuous, then there exists a change of basis Ai on Wi such that
where , and
for all . If
, then
In both cases is a commuting family of matrices on Wi satisfying Cauchy’s functional equation. Furthermore, if v is a common eigenvector of
such that
, then ν is linear in x.
Proof.
Assume first that ηi, the associated solution of Cauchy’s functional equation, is non-continuous. For each Wi we have the decomposition where Vj are subspaces from the SRPD. Since ηi is non-continuous, each Vj in the direct sum has to be of second type, hence by (Equation4.3
(4.3)
(4.3) ) on each Vj we have
By definition of ηi we have and hence there exists
such that
. Let
with
where Cj is of dimension . Then, we have
Set . Then
satisfy
By construction the common eigenvalues of satisfy
. If
, then by Theorem 4.3 there exists
such that
. By the definition of Wi, it follows that the imaginary parts of the eigenvalues of
have to be equivalent to
, hence they have to be linear.
Theorem 5.3.
Let g be a non-degenerate semigroup acting on V which satisfies Assumption 2.5. Then, there exists a matrix M and a semigroup , where SO(d) is the set of special orthogonal matrices, such that
with M commuting with S(x).
Proof.
Consider the SRPD from Theorem 3.2, and define
. Clearly,
. For each eigenvalue
on Vi, we have
Since , we have
. Moreover,
, and so
for
and
. It follows that µi is locally bounded and hence that
for some
. As in the proof of Theorem 4.3, on Vi we have
where
is diagonalizable with eigenvalues
, and thus
As in Theorem 4.3, set
Then, since , we obtain
where F(x) is again locally bounded. Since the operator norm in finite dimensions is equivalent to the -norm, it follows that each entry of
is locally bounded and satisfies Cauchy’s functional equation. Thus, there exists a nilpotent linear map Pi such that
. Let
be the partitioned SRPD of V. Assume first that the solution associated with Wi is
. Since
we have
. The real part of the eigenvalues of
is linear in x and by Theorem 5.2 also the complex parts have to be linear since
. Since
is diagonalizable and all its eigenvalues are continuous in x,
is continuous in x and hence
. Setting
and
yields
. By Theorem 5.2 and (Equation5.1
(5.1)
(5.1) ), we have for Wi with ηi non-continuous
with
Hence, setting , we have
. Thus
Next we show that is an isometry on Wi. Since
, we have
for any . Hence, for
we have
Fix an arbitrary , and set
. The graph of ηi is dense in
(see Appendix A), and so there is a sequence of positive reals xn with
such that
. Then, for
we obtain
where since f is right-continuous at 0. Choose
such that
and
. Then, since
, we obtain
since . Hence, we obtain
which implies that is an isometry on Wi. Next we show that all Wi are pairwise orthogonal. Let
and
for i ≠ j. By Lemma A.3, w.l.o.g. there is a sequence
such that
,
and
. Hence
and
with
. We have the identity
. Applying this to
we obtain
Hence we have . Set
. Then
, and applying (Equation5.3
(5.3)
(5.3) ) to
and taking the limit along
yields
where the last equality follows from the fact that each is an isometry on Wi and Wj. Since the inequality above also holds for
, we obtain the equality
Since were arbitrary, this shows orthogonality. Hence, the decomposition
is orthogonal, and since
is an isometry on Wi, it follows that
is in SO(d). Setting
, we obtain
Corollary 5.4.
Assume that is a semigroup with A being an invertible matrix such that S(x) is an isometry for each
. Then, there exists an orthogonal matrix U such that
Proof.
One can easily verify the identities and
Choose x > 0 such that . Such an x clearly exists since ν is linear on
. Choose any
of unit length and set
The definition of u is invariant under the choice of x as long as . This can be seen by noticing that
Hence we obtain . We have
where the last equality follows from
Similarly, we can show that u is also of unit length. Set . Clearly, H is invariant under
and hence so is
since each
. In this manner, we can construct an orthonormal basis, and we denote by U the matrix associated with this change of basis. Then, we have
Lemma 5.5.
Suppose that the semigroup g, acting on V, satisfies Assumption 2.5. Let be the decomposition of Corollary 3.8 such that
is invertible. Then, for any
we have
Proof.
By Theorem 5.3, we have . Since
, we obtain
which is continuous in x. Hence
Corollary 5.6.
Suppose that the semigroup g, acting on V, satisfies Assumption 2.5. Then, the decomposition from Corollary 3.8 is orthogonal.
Proof.
Let with
being non-degenerate and
. Assume V1 is not orthogonal to V2. Then, there exists
such that
, where
denotes the orthogonal projection onto V2. By Lemma 5.5, we have
Calculating the same limit for we obtain
Since and
, we have
. Hence
but this contradicts . Hence
.
Proof.
Proof of Theorem 2.6
By Corollary 5.6, we have the orthogonal decomposition with
and
non-degenerate. Applying Theorem 5.3 to
yields the result.
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References
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Appendix A.
Cauchy’s functional equation
It is classical that all continuous solutions of the EquationEquation (1.3)(1.3)
(1.3) ,
, are linear, and that the non-linear solutions are not continuous, even not Lebesgue measurable, and have dense graphs. For this, and further references, we refer to [2, Section 1.1]. In this section, we provide two auxiliary results on Cauchy’s equation. They concern lifting solutions from an interval to the real line, resp. the joint behavior of two solutions that differ by a non-linear function.
Lemma A.1.
Let be a solution to Cauchy’s functional EquationEquation (1.3)
(1.3)
(1.3) on
with a > 0. Then, there exists a solution
of (Equation1.3
(1.3)
(1.3) ) such that
. The solution
is linear if and only if f is linear.
Proof.
Take a Hamel basis of
such that
for every
. This is clearly possible by rescaling every basis element if necessary. For any
there exists a finite subset
and
such that
. The function
satisfies
. If f is linear, then clearly
is linear as well. If f is not linear then there exist two basis elements r1 and r2 such that
, and hence
is also not linear.
Definition A.2.
We say that two solutions ν and η of Cauchy’s functional equation are equivalent if is linear.
Lemma A.3.
Let be two non-equivalent solutions of Cauchy’s functional equation. Then there exists a sequence
in
, converging to 0, such that either
and
with
or vice versa.
Proof.
Choose such that the two vectors
and
are linearly independent and
. This is possible, since
for all would imply that f − g is linear. Assume w.l.o.g. that
. Since f and g are both linear on
and
and v1 and v2 are linearly independent, there exist sequences
and
such that
for every n,
and
. We show that
has the desired property. Clearly
and
Hence