EMBEDDING OPERATORS INTO STRONGLY CONTINUOUS ...

EMBEDDING OPERATORS INTO C 0 -SEMIGROUPS 6
Proof. If 0 < dim(kerP) < ∞, P is not embeddable by Theorem 3.1. The rest follows from the
decomposition H = ker P ⊕ rg P into two invariant subspaces, Lemma 4.6 and embeddability of
the identity operator (on any Banach space) into the identity semigroup.
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We finish this section with the following natural question.
Open Problem 4.8. Does the necessary condition given in Theorem 3.1 suffice to embed partial
isometries on Hilbert spaces?
Note that for a partial isometry T the condition is sufficient in each of the following cases:
(a) T is injective, i.e., T is an isometry (Theorem 4.4);
(b) T is surjective, i.e., T ∗ is an isometry (Corollary 4.5);
(c) ker T reduces T.
The argument in (c) is similar to the proof of Proposition 4.7 using Theorem 4.4.
5. More examples, a category result
In this section we give more examples of embeddable operators.
The following result extends the argument from (the proof of) Proposition 4.1 to normal
operators.
Proposition 5.1. Let T be a normal operator on a Hilbert space. Then T is embeddable if and
only if T is injective or dim kerT = ∞.
Proof. By the spectral theorem, see e.g. Halmos [13], we may assume that T is a multiplication
operator on H = L 2 (Ω, µ) for Ω = σ(T) and some Borel measure µ, say Tf = mf with
m ∈ L ∞ (Ω, µ). Note that the essential image of m equals σ(T).
Assume first that T is injective, whence µ(m −1 (0)) = 0. Using the principal value of the
logarithm on C \ {0}, which is measurable, define (T(t)) t≥0 by
T(t)f := e t log m f, f ∈ L 2 (Ω, µ), t ≥ 0.
Each T(t) is a bounded operator on L 2 (Ω, µ). Moreover, the family (T(t)) t≥0 is strongly continuous
by Lebesgue’s dominated convergence theorem. The semigroup law and the property
T(1) = T are clear, so T is embeddable.
Assume now dim ker T = ∞. Since T is normal, we have rg T ∗ = (kerT) ⊥ = (kerT ∗ ) ⊥ = rg T.
Therefore ker T and rg T reduce T and H = kerT ⊕ rg T. The restriction T | ker T is embeddable
by Lemma 4.6. On the other hand, T | rg T
is an injective normal operator and hence embeddable
by the first part of the proof. This shows that T is embeddable as well.
The remaining case is answered in the negative by Theorem 3.1.
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Remark 5.2. The above proof shows that T can be embedded into a normal C 0 -semigroup
whenever T is injective. Moreover, if T is invertible, then 0 has positive distance to σ(T) and T
can be embedded into a normal C 0 -group with bounded generator employing the same definition
for t ∈ R (cf. Proposition 4.1).
Remark 5.3. Clearly, also each operator T similar to a normal operator is embeddable if and
only if T is injective or dim kerT = ∞. For example, Sz.-Nagy [19] showed that a bijective
operator T on a Hilbert space satisfying sup j∈Z ‖T j ‖ < ∞ is similar to a unitary operator and
hence embeddable into a C 0 -group by Proposition 4.1. We refer also to van Casteren [2] for