From Classical Mechanics to Quantum Field Theory

94 From Classical Mechanics to Quantum Field Theory. A Tutorial

Definition 2.2.20.LetAbe an operator in the complex Hilbert spaceH.

(1)Ais said to beclosedif thegraphofA, that is the set

G(A):={(x, Ax)⊂H×H|x∈D(A)},

is closed in the product topology ofH×H.

(2)Aisclosableif it admits extensions in terms of closed operators. This is

equivalent to saying that the closure of the graph ofAis the graph of an

operator, denoted byA, and called theclosureofA.

(3)IfAis closable, a subspaceS⊂D(A)is calledcoreforAifA|S=A.

Referring to (2), we observe that, given an operatorA, we can always define the
closure of the graphG(A)inH×H. In general, this closure is not the graph of an
operator because there could exist sequencesD(A)xn→xandD(A)x′n→x
such thatTxn→yandTxn→y′withy=y′. However, both pairs (x, y)and
(x, y′)belongtoG(A). If this case does not take place – and this is equivalent
to condition (a) below whenmaking use of linearity –G(A) is the graph of an
operator, denoted byA, that is closed by definition. ThereforeAadmits closed
operatorial extensions: at leastA. If, conversely,Aadmits extensions in terms
of closed operators, the intersection of the (closed) graphs of all these extensions
G(A) is still closed and it is still the graph of an operator again, which must
councide withAby definition.

Remark 2.2.21.
(a)Directly from the definition and using linearity,Ais closable if and only
if there are no sequences of elementsxn∈D(A)such thatxn→ 0 andAxn→y
withy=0asn→+∞.SinceG(A)is the union ofG(A)and its accumulation
points inH×Hand, ifAis closable, it is also the graph of the operatorA,we
conclude that

(i)D(A)is made of the elementsx∈Hsuch thatxn→xandAxn→yxfor

some sequences{xn}n∈N⊂D(A)and someyx∈D(A)and

(ii)Ax=yx.

(b)As a consequence of (a) one has that, ifAis closable, thenaA+bIis
closable andaA+bI=aA+bIfor everya, b∈C. N.B.This result generally
fails if replacingIfor some closable operatorB.
(c)Directly from the definition,Ais closed if and only ifD(A)xn→x∈H
andAxn→y∈Himply bothx∈D(A)andy=Ax.

A useful porposition is the following.