Download One-parameter Semigroups of Positive Operators by Wolfgang Arendt, Annette Grabosch, Günther Greiner, Ulrich PDF

By Wolfgang Arendt, Annette Grabosch, Günther Greiner, Ulrich Groh, Heinrich P. Lotz, Ulrich Moustakas, Rainer Nagel, Frank Neubrander, Ulf Schlotterbeck

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In this case for each integer k ≥ 1/s1 there exists an element in Zk ∈ B1/k (0) such that exp(Zk ) ∈ H and Zk ∈ / Lie(H). We write Zk = Xk +Yk with Xk ∈ Lie(H) and 0 = Yk ∈ K. Then ϕ(Zk ) = exp(Xk ) exp(Yk ) . Since exp(Xk ) ∈ H, we see that exp(Yk ) ∈ H. We also observe that Yk ≤ 1/k. Let εk = Yk . Then 0 < εk ≤ 1/k ≤ s1 . For each k there exists a positive integer mk such that s1 ≤ mk εk < 2s1 . Hence s1 ≤ mkYk < 2s1 . 21) Since the sequence mkYk is bounded, we can replace it with a subsequence that converges.

It follows that B([X,Y ]v, w) = −B(v, [X,Y ]w), and hence so(V, B) is a Lie subalgebra of gl(V ). Suppose V is finite-dimensional. Fix a basis {v1 , . . , vn } for V and let Γ be the n×n matrix with entries Γi j = B(vi , v j ). 2, we see that T ∈ so(V, B) if and only if its matrix A relative to this basis satisfies At Γ + Γ A = 0 . 8) can be written as At = −Γ AΓ −1 . In particular, this implies that tr(T ) = 0 for all T ∈ so(V, B). Orthogonal Lie Algebras Take V = Fn and the bilinear form B with matrix Γ = In relative to the standard basis for Fn .

It is a fundamental result in Lie theory that all homomorphisms from R to GL(n, R) are obtained in this way. 5. Let ϕ : R additive group R to GL(n, R). Then there exists a unique X ∈ Mn (R) such that ϕ(t) = exp(tX) for all t ∈ R. Proof. The uniqueness of X is immediate, since d exp(tX) dt t=0 =X . To prove the existence of X, let ε > 0 and set ϕε (t) = ϕ(εt). Then ϕε is also a continuous homomorphism of R into GL(n, R). 3 we can choose ε such that ϕε (t) ∈ exp Br (0) for |t| < 2, where r = (1/2) log 2.

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