Nuclear norm

trace norm

A norm on the space $ N ( X, Y) $ of nuclear operators (cf. Nuclear operator) mapping a Banach space $ X $ into a Banach space $ Y $.

Let $ X $ and $ Y $ be Banach spaces over the field of real or complex numbers, let $ L ( X, Y) $ be the space of all continuous linear operators mapping $ X $ into $ Y $, and let $ F ( X, Y) $ be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). The Banach dual of $ X $ is denoted by $ X ^ \prime $, and the value of a functional $ x ^ \prime \in X ^ \prime $ at a vector $ x \in X $ by $ \langle x, x ^ \prime \rangle $.

Every nuclear operator $ A \in N ( X, Y) $ can be represented in the form

$$ \tag{1 } x \mapsto Ax = \ \sum _ {i = 1 } ^ \infty \langle x, x _ {i} ^ \prime \rangle y _ {i} , $$

where $ \{ x _ {i} ^ \prime \} $ and $ \{ y _ {i} \} $ are sequences in $ X ^ \prime $ and $ Y $, respectively, such that

$$ \sum _ {i = 1 } ^ \infty \| x _ {i} ^ \prime \| \| y _ {i} \| < \infty ; $$

such representations are called nuclear. The quantity

$$ \tag{2 } \| A \| _ {1} = \ \inf \ \sum _ {i = 1 } ^ \infty \| x _ {i} ^ \prime \| \| y _ {i} \| , $$

where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of $ A $. The space $ N ( X, Y) $ with this norm is a Banach space that contains $ F ( X, Y) $ as a dense linear subspace. If $ A \in N ( X, Y) $, then the adjoint operator $ A ^ \prime $ belongs to $ N ( Y ^ \prime , X ^ \prime ) $, and $ \| A ^ \prime \| _ {1} \leq \| A \| _ {1} $. Let $ \| \cdot \| $ denote the usual operator norm in $ L ( X, Y) $. Then $ \| A \| \leq \| A \| _ {1} $ for all $ A \in N ( X, Y) $. If $ A \in L ( Y, Z) $ and $ B \in N ( X, Y) $, then $ AB \in N ( X, Z) $, and $ \| AB \| _ {1} \leq \| A \| \| B \| _ {1} $; if $ A \in N ( Y, Z) $ and $ B \in L ( X, Y) $, then $ AB \in N ( X, Z) $, and $ \| AB \| _ {1} \leq \| A \| _ {1} \| B \| $. Any operator $ F \in F ( X, Y) $ can be represented in the form

$$ \tag{3 } x \mapsto Fx = \ \sum _ {i = 1 } ^ { n } \langle x, x _ {i} ^ \prime \rangle y _ {i} . $$

The quantity

$$ \tag{4 } \| F \| _ {1} ^ {0} = \ \inf \ \sum _ {i = 1 } ^ { n } \| x _ {i} ^ \prime \| \| y _ {i} \| , $$

where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of $ F $. The space $ F ( X, Y) $ can be identified with the tensor product $ X ^ \prime \otimes Y $. Here, to an operator $ F $ of the form (3) there corresponds the element

$$ \tag{5 } u = \ \sum _ {i = 1 } ^ { n } x _ {i} ^ \prime \otimes y _ {i} \in \ X ^ \prime \otimes Y, $$

and the finite nuclear norm (4) goes into the norm

$$ \tag{6 } \| u \| = \ \inf \ \sum _ {i = 1 } ^ { n } \| x _ {i} ^ \prime \| \| y _ {i} \| , $$

where the infimum is taken over all finite representations of $ u $ in the form (5). This norm is called the tensor (or cross) product of the norms in $ Y $ and in $ X ^ \prime $. The completion of $ X ^ \prime \otimes Y $ with respect to the norm (6) is denoted by $ X ^ \prime \widehat \otimes Y $. The mapping $ X ^ \prime \otimes Y \rightarrow L ( X, Y) $, under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator $ \Gamma : X ^ \prime \widehat \otimes Y \rightarrow L ( X, Y) $. The range of $ \Gamma $ is $ N ( X, Y) $. If $ \Gamma $ establishes a one-to-one correspondence between $ X ^ \prime \widehat \otimes Y $ and $ N ( X, Y) $, then $ N ( X, Y) $ coincides with the closure of $ F ( X, Y) $ with respect to the norm (4); in this case the restriction of the nuclear norm to $ F ( X, Y) $ is the same as the finite nuclear norm. But, in general, $ \Gamma $ may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in $ X ^ \prime \widehat \otimes Y $( see Nuclear operator).

Let $ X = Y = H $, where $ H $ is a separable Hilbert space, let $ L ( H) = L ( H, H) $ be the algebra of bounded operators on $ H $, and let $ L _ {1} ( H) = N ( H, H) $ be the ideal of nuclear operators in $ L ( H) $. In this case $ \Gamma $ is one-to-one, for operators of finite rank the nuclear norm coincides with the finite nuclear norm, and each $ A \in L _ {1} ( H) $ has a trace $ \mathop{\rm tr} A $( see Nuclear operator). The nuclear norm of an operator $ A \in L _ {1} ( H) $ coincides with $ \mathop{\rm tr} [( A ^ {*} A) ^ {1/2} ] $, where $ A ^ {*} $ is the adjoint of $ A $ in $ H $. The nuclear norm is connected with the Hilbert–Schmidt norm $ \| \cdot \| _ {2} $ by $ \| A \| _ {2} \leq \| A \| _ {1} $. The general form of a continuous linear functional on the Banach space $ L _ {1} ( H) $ is given by

$$ \tag{7 } A \rightarrow \mathop{\rm tr} AB, $$

where $ B $ is an arbitrary operator from $ L ( H) $, and the norm of the functional (7) coincides with $ \| B \| $. Consequently, $ L ( H) $ is isometric to the dual of $ L _ {1} ( H) $. Formula (7) also gives the general form of a linear functional on the closed subspace $ L _ \infty ( H) $ of $ L ( H) $ that consists of all completely-continuous (compact) operators; here $ A \in L _ \infty ( H) $ and $ B $ ranges over $ L _ {1} ( H) $. In this case the norm of the functional (7) coincides with $ \| B \| _ {1} $, that is, the space $ L _ {1} ( H) $ of nuclear operators with the nuclear norm is isometric to the dual of $ L _ \infty ( H) $ in the usual operator norm. These results have non-trivial generalizations to the case of operators on Banach spaces.

Example. Let $ X = Y = l _ {1} $ be the space of summable sequences. An operator $ A \in L ( l _ {1} , l _ {1} ) $ is contained in $ N ( l _ {1} , l _ {1} ) $ if and only if there is an infinite matrix $ ( \sigma _ {ik} ) $ such that $ A $ sends $ \{ \xi _ {k} \} \in l _ {1} $ to $ \{ \eta _ {i} \} = \{ \sum _ {k = 1 } ^ \infty \sigma _ {ik} \xi _ {k} \} \in l _ {1} $, and $ \sum _ {i = 1 } ^ \infty \sup _ {k} | \sigma _ {ik} | < \infty $. In this case, $ \| A \| _ {1} = \sum _ {i = 1 } ^ \infty \sup _ {k} | \sigma _ {ik} | $.

References

[1]	A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955)
[2]	A. Pietsch, "Operator ideals" , North-Holland (1980)
[3]	A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German)
[4]	I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian)
[5]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)
[6]	K. Maurin, "Methods of Hilbert spaces" , PWN (1967)
[7]	M.M. Day, "Normed linear spaces" , Springer (1958)

References

[a1]	A. Pietsch, "Eigenvalues and -numbers" , Cambridge Univ. Press (1987)
[a2]	A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques" Bol. Soc. Mat. São Paulo , 8 (1956) pp. 1–79
[a3]	H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)