Let and be Banach spaces over the field of real or complex numbers, let be the space of all continuous linear operators mapping into , and let be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). The Banach dual of is denoted by , and the value of a functional at a vector by .
Every nuclear operator can be represented in the form
where and are sequences in and , respectively, such that
such representations are called nuclear. The quantity
where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of . The space with this norm is a Banach space that contains as a dense linear subspace. If , then the adjoint operator belongs to , and . Let denote the usual operator norm in . Then for all . If and , then , and ; if and , then , and . Any operator can be represented in the form
where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of . The space can be identified with the tensor product . Here, to an operator of the form (3) there corresponds the element
and the finite nuclear norm (4) goes into the norm
where the infimum is taken over all finite representations of in the form (5). This norm is called the tensor (or cross) product of the norms in and in . The completion of with respect to the norm (6) is denoted by . The mapping , under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator . The range of is . If establishes a one-to-one correspondence between and , then coincides with the closure of with respect to the norm (4); in this case the restriction of the nuclear norm to is the same as the finite nuclear norm. But, in general, may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in (see Nuclear operator).
Let , where is a separable Hilbert space, let be the algebra of bounded operators on , and let be the ideal of nuclear operators in . In this case is one-to-one, for operators of finite rank the nuclear norm coincides with the finite nuclear norm, and each has a trace (see Nuclear operator). The nuclear norm of an operator coincides with , where is the adjoint of in . The nuclear norm is connected with the Hilbert–Schmidt norm by . The general form of a continuous linear functional on the Banach space is given by
where is an arbitrary operator from , and the norm of the functional (7) coincides with . Consequently, is isometric to the dual of . Formula (7) also gives the general form of a linear functional on the closed subspace of that consists of all completely-continuous (compact) operators; here and ranges over . In this case the norm of the functional (7) coincides with , that is, the space of nuclear operators with the nuclear norm is isometric to the dual of in the usual operator norm. These results have non-trivial generalizations to the case of operators on Banach spaces.
Example. Let be the space of summable sequences. An operator is contained in if and only if there is an infinite matrix such that sends to , and . In this case, .
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Nuclear norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nuclear_norm&oldid=19242