1 The transfer-semigroup realization
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Let \(X\) be a finite-dimensional real or complex vector space, let \(A\in\operatorname{End}(X)\), and let \(s\in X\) be a primitive seed. The basic continuous transfer is
while a positive operator \(L\) is more naturally written
Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with \(e^{tA}\); positive diffusion and elliptic operators are usually written with \(e^{-tL}\).
The common operator chain is
Definition 1.1: Transfer-semigroup seed system
A finite transfer-semigroup seed system is a tuple
where \(X\) is the state space, \(s\) is the seed, \(A\) is a transfer generator, the \(P_\alpha\) are pairwise orthogonal idempotents summing to the identity, and \(\tau\) specifies the topology in which a completion or limit is interpreted.
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Theorem 1.2: Finite semigroup law
For every matrix \(A\) and all \(s,t\in{\mathbb R}\),
Moreover, \(T(0)=I\) and \(T'(t)=AT(t)=T(t)A\).
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Proof of Theorem 1.2
Both exponentials are absolutely convergent matrix power series. Since every power of \(A\) commutes with every other power, the Cauchy product gives
Termwise differentiation is valid in finite dimension and gives the derivative identities.
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Boundary 1.3: Infinite-dimensional generators
For an unbounded operator, writing \(e^{tA}\) is not a definition unless a semigroup-generation theorem applies. The finite matrix identities in this paper remain algebraically correct models, but a continuum realization must state the operator domain, closure, growth bound, and topology of convergence.
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