A bounded linear operator \(T\) on a Banach space \(X\) is called hyperbolic if there is a splitting \(X=X_s \oplus X_u\), where \(X_s\) and \(X_u\) are closed \(T\)-invariant subspaces of \(X\), the restriction \(T|X_u\) is invertible, and \(\|(T|X_s)^n\| \leq 1/2\) and \(\|(T|X_u)^{-n}\| \leq 1/2\) for some \(n \in \mathbb{N}\). The operator \(T\) is hyperbolic if and only if the spectrum \(\sigma(T)\) does not meet the unit circle \(\mathbb{T}\).
The aim of this paper is to introduce and study quasi-hyperbolic operators and \(C_0\)-semigroups. An operator \(T\) on \(X\) is said to be quasi-hyperbolic if there is \(n \in \mathbb{N}\) such that \(\max(\|T^{2n} x\|, \|x\|) \geq 2 \|T^n x\|\) for each \(x \in X\). Every hyperbolic operator is quasi-hyperbolic. In Section 2, quasi-hyperbolic operators are identified as restrictions of hyperbolic operators to invariant subspaces. A quasi-hyperbolic operator can be characterized also by the simple spectral property that the approximate spectrum \(\sigma_{ap}(T)\) does not meet the unit circle. These characterizations depend on a deep extension theorem due to C.J.Read [Trans.Am.Math.Soc.308, No.1, 413–429 (1988; Zbl 0663.47006)].
Many interesting examples are presented. In Section 3, the authors characterize quasi-hyperbolic \(C_0\)-semigroups in terms of their generator and in terms of lower Fourier multipliers. In particular, the quasi-hyperbolicity of a \(C_0\)-semigroup \((T(t)\) on a Hilbert space with generator \(A\) is equivalent to the lower bound \(\|(A-is)x\| \geq c \|x\|\) for all \(s \in \mathbb{R}\) and all \(x\) in the domain of \(A\). However, this condition does not characterize quasi-hyperbolic \(C_0\)-semigroups on general Banach spaces as an example shows. In the final section, it is shown that the lower bound ensures that nonzero complete trajectories \((T(t))_{t \geq 0}\) grow faster than polynomially in a pointwise sense and grow exponentially in an integral norm.