Sentence examples for maximal sectorial in from inspiring English sources

We also show that the operator A is maximal sectorial in (overline{D(A)}) if and only if (mathcal{T}) is maximal sectorial.

We say that A is maximal sectorial in (overline{D(A)}) if it has no proper sectorial extension (A_{1}) in (overline{D(A)}).

Hence if (mathcal{T}) is a maximal sectorial linear relation in ℋ, then the operator A is maximal sectorial in (overline{D mathcal{T})}). Now let us assume that an operator A with the stated properties does exist and suppose that (mathcal{T}) is not maximal sectorial in ℋ.

(ii) (mathcal{T}) is maximal sectorial if and only if the operator A is maximal sectorial in (overline{D mathcal{T})}) and mathcal{T}(x) = mathcal{T}(0) + Ax (4.18) for all (xin D mathcal{T})=D(A)).  .

As in the case of sectorial operators, we say that (mathcal{T}) is maximal sectorial in ℋ if there does not exist a sectorial linear relation (mathcal{T}_{1}) in ℋ such that (G(mathcal{T})subset G(mathcal{T}_{1})).

(mathcal{T}) is maximal sectorial if and only if the operator A is maximal sectorial in (overline{D mathcal{T})}) and mathcal{T}(x) = mathcal{T}(0) + Ax (4.18) for all (xin D mathcal{T})=D(A)). (i) Let (mathcal{T}) be a sectorial linear relation in ℋ with domain (D mathcal{T})) and decompose ℋ as mathcal{H} = overline{D mathcal{T})}^{perp}oplus overline {D mathcal{T})}.

Hence B is a sectorial extension of A. This contradicts the maximality of A. This contradiction implies that (mathcal{T}) is a maximal sectorial linear relation in ℋ. □.

(ii) Now assume that (mathcal{T}) is a maximal sectorial linear relation in ℋ with domain (D mathcal{T})).

Since t is densely defined in (overline{D t)}), Theorem 2.4(i) implies that there exists a maximal sectorial operator A in (overline {D t)}) such that (D(A) subset D t)) and (t u,v =langle Au, vrangle) for every (uin D(A)) and every (vin D t)).

There exists a maximal sectorial operator A in ℋ such that (i) (D(A) subset D t)) and t u,v) = langle Au, vrangle for every (uin D(A)) and (vin D t));   (ii) (D(A)) is a core of t;   (iii) if (uin D t)), (winmathcal{H}) and t u,v =langle w,vrangle holds for every v belonging to a core of t, then (uin D(A)) and (Au=w).

Maximal sectorial operators are useful in operator representations of sectorial forms as seen from the following theorem which is referred to in the literature as Kato's first representation theorem.