Theorem 12 Let m x and M x, m x ≤ M x, be the bounds of the operator x = ∫ T ϕ t ( x t ) d μ ( t ).
Corollary 7 Let ( x t ) t ∈ T be a field of strictly positive operators, let m x and M x, m x ≤ M x, be the bounds of the operator x = ∫ T ϕ t ( x t ) d μ ( t ).
Theorem 5 Let m x and M x, m x ≤ M x, be the bounds of the operator x = ∫ T ϕ t ( x t ) d μ ( t ) and let m x ˜ be the lower bound of the operator x ˜.
Theorem 8 Let m x and M x, m x ≤ M x, be the bounds of the operator x = ∫ T ϕ t ( x t ) d μ ( t ) and let f : [ m, M ] → R be a continuous function.
In this paper, the bounds of the operator H β from L p to L q and from L 1 to L n n − β, ∞ are explicitly worked out.
Let ( m x, M x ) ∩ [ m t, M t ] = ∅, t ∈ T, and a < b, where m x and M x, m x ≤ M x, are the bounds of the operator x = ∫ T ϕ t ( x t ) d μ ( t ) and a = sup { M t : M t ≤ m x, t ∈ T }, b = inf { m t : m t ≥ M x, t ∈ T }.
Part III is devoted to perturbation bounds on the operator norm of semigroups and a new (short) proof of the off-diagonal estimates of the heat kernel associated with a divergence operator.
Let (m_{x}), (M_{x}) and (n_{y}), (N_{y}) be bounds of the operators (Phi(X)) and (Psi(Y)), respectively (see Figure 3).
Further, we obtain the upper bound of the operator norm (|A_{k}|_) which implies the continuity of the Hardy-Knopp operator between two different Orlicz spaces.
end{aligned} Here, (Vert cdot Vert _{W^{1,q}(Omega)}) denotes the σ-weighted (W^{1,q}) norm (2) for given (sigma>0), and (A_{q} (Omega )) is the upper bound of the operator norm derived by Theorem 4.2 with (gamma=sigma^{1/q}).
This means that for actual implementation of CQ algorithm (1.2), one has first to know at least an upper bound of the operator (matrix) norm ∥ A ∥, which is in general difficult.
