Fact-checked by Grok 2 weeks ago

Operator norm

In mathematics, particularly in functional analysis, the operator norm is a measure of the "size" or magnitude of a bounded linear operator T: X \to Y between normed vector spaces X and Y, defined as \|T\| = \sup \{ \|Tx\|_Y : x \in X, \|x\|_X \leq 1 \}, which is equivalently the infimum of all constants M \geq 0 such that \|Tx\|_Y \leq M \|x\|_X for all x \in X.^[1]^[2] This norm captures the supremum stretching factor of the operator on the unit ball of X, and a linear operator admits an operator norm if and only if it is bounded, which is equivalent to being continuous.^[3]^[2] The operator norm endows the space B(X, Y) of all bounded linear operators from X to Y with the structure of a normed vector space, and if Y is a Banach space, then B(X, Y) becomes a Banach space itself under this norm.^[2]^[3] Key properties include homogeneity (\|cT\| = |c| \|T\| for scalars c), non-negativity (\|T\| = 0 if and only if T = 0), and submultiplicativity (\|ST\| \leq \|S\| \|T\| for composable operators S and T), making it compatible with the algebraic structure of operator composition.^[1]^[3] In Hilbert spaces, additional characterizations arise, such as for self-adjoint operators where \|T\| = \sup \{ |\langle Tx, x \rangle| : \|x\| = 1 \}, linking the norm to quadratic forms.^[3] Operator norms play a central role in spectral theory, where the spectral radius satisfies r(T) = \lim_{n \to \infty} \|T^n\|^{1/n} \leq \|T\|, providing bounds on eigenvalues and invertibility criteria, such as the Neumann series for \|T\| < 1 yielding (I - T)^{-1} = \sum_{n=0}^\infty T^n.^[3] They are also essential in the study of compact operators, adjoints (with \|T^*\| = \|T\| in Hilbert spaces), and semigroups of operators, where growth bounds like \|S(t)\| \leq M e^{\omega t} ensure stability and well-posedness of evolution equations.^[2]^[3] These features underpin theorems like the uniform boundedness principle and open mapping theorem, facilitating the analysis of infinite-dimensional phenomena in applications from partial differential equations to quantum mechanics.^[3]

Fundamentals

Definition

In functional analysis, the operator norm of a bounded linear operator T: X \to Y between normed vector spaces (X, \|\cdot\|_X) and (Y, \|\cdot\|_Y) is defined as

\|T\| = \sup \left\{ \frac{\|Tx\|_Y}{\|x\|_X} : x \in X, \, x \neq 0 \right\}.

^[4]^[5] This quantity represents the maximum factor by which T can amplify the norm of input vectors from X, providing a measure of the operator's "size" or sensitivity to inputs.^[4]^[5] An equivalent formulation is

\|T\| = \sup \left\{ \|Tx\|_Y : x \in X, \, \|x\|_X \leq 1 \right\},

which is the least upper bound of the norms of images of vectors in the closed unit ball of X.^[4]^[5] To see the equivalence, note that the homogeneity of norms (\|\lambda z\| = |\lambda| \|z\| for scalars \lambda) implies that for any x \neq 0, the unit vector u = x / \|x\|_X satisfies \|Tu\|_Y = \|Tx\|_Y / \|x\|_X, so the supremum over ratios equals the supremum over the unit ball (or precisely the unit sphere, as the maximum on the boundary extends to the ball by subadditivity).^[4]^[5] The operator T is bounded if and only if \|T\| < \infty, meaning there exists a constant C = \|T\| such that \|Tx\|_Y \leq \|T\| \|x\|_X for all x \in X.^[4]^[5] The collection of all such bounded operators, denoted B(X, Y), forms a normed vector space under pointwise addition and scalar multiplication, equipped with the operator norm \|\cdot\|.^[4]^[5]

Motivation

In the context of normed vector spaces, linear operators play a central role in functional analysis, where continuity of such operators is equivalent to boundedness.^[6] This equivalence underscores the importance of quantifying the "boundedness" of a linear map from one normed space to another, providing a foundation for studying how these maps preserve or distort the structure of vectors under the given norms.^[6] The concept of the operator norm emerged as a natural generalization of matrix norms from finite-dimensional spaces to infinite-dimensional settings, driven by the need to handle operators on spaces like those of continuous functions or integrable functions.^[7] This development originated in the early 20th-century work of Stefan Banach, who in his 1920 doctoral thesis introduced complete normed linear spaces—now known as Banach spaces—and laid the groundwork for operator theory through publications in the 1920s, culminating in his 1932 monograph Théorie des opérations linéaires.^[7] Banach's motivation stemmed from applications in solving integral equations and spectral problems, where finite-dimensional tools proved insufficient for infinite-dimensional phenomena.^[7] Operator norms are essential for endowing the space of bounded linear operators between Banach spaces with a natural topology, which facilitates the analysis of convergence, approximation, and compactness in operator sequences.^[7] Without such a norm, studying the behavior of operators in infinite dimensions—such as their limits or compositions—would lack the metric structure needed for rigorous proofs and applications in areas like partial differential equations and quantum mechanics.^[7] Intuitively, the operator norm captures the "size" of a linear operator by measuring its maximum amplification effect on vectors, much like a vector norm quantifies length in the space.^[8] This stretching factor provides a scalar benchmark for the operator's influence, enabling comparisons and bounds in theoretical and computational contexts.^[8]

Induced Norms

General induced norms

The induced norm on a linear operator between normed vector spaces arises naturally from the vector norms on the domain and codomain, providing a measure of the operator's "amplification" effect relative to those norms. Consider normed vector spaces X and Y over the same field (typically \mathbb{R} or \mathbb{C}), equipped with vector norms \|\cdot\|_\alpha on X and \|\cdot\|_\beta on Y. For a linear operator T: X \to Y, the (\alpha, \beta)-induced norm, also called the subordinate norm, is defined as

\|T\|_{\alpha,\beta} = \sup\left\{ \frac{\|Tx\|_\beta}{\|x\|_\alpha} \;\middle|\; x \in X \setminus \{0\} \right\}.

^[9] This supremum is always finite if T is bounded, and it equals zero if and only if T is the zero operator. An equivalent formulation is

\|T\|_{\alpha,\beta} = \sup\left\{ \|Tx\|_\beta \;\middle|\; x \in X, \; \|x\|_\alpha \leq 1 \right\}, $$ which emphasizes the maximum stretch of the unit ball in $X$ under $T$, measured in the $Y$-norm.[](https://www.cis.upenn.edu/~cis5150/cis515-20-sl4.pdf) A fundamental property of the induced norm is its compatibility with the underlying vector norms: for all $x \in X$,

|Tx|\beta \leq |T|{\alpha,\beta} |x|_\alpha.

[](https://www.maths.usyd.edu.au/u/athomas/FunctionalAnalysis/daners-functional-analysis-2017.pdf) This inequality follows directly from the definition, as the ratio $\|Tx\|_\beta / \|x\|_\alpha$ is bounded above by $\|T\|_{\alpha,\beta}$ for $x \neq 0$, and it holds trivially for $x = 0$. The induced norm thus quantifies the boundedness of $T$ in a way that aligns precisely with the geometry of the normed spaces involved. Every such induced norm satisfies the axioms of a norm on the space of bounded linear operators from $X$ to $Y$ (including submultiplicativity: $\|ST\|_{\alpha,\gamma} \leq \|S\|_{\beta,\gamma} \|T\|_{\alpha,\beta}$ for compatible norms and a suitable intermediate space), making it an operator norm. However, not every operator norm (i.e., every submultiplicative norm on the operator space) is induced by vector norms on the domain and codomain in this manner. Induced norms are classified as consistent or mixed depending on the relationship between the vector norms. A consistent induced norm occurs when the same vector norm is used on both domain and codomain (i.e., $\alpha = \beta$), often simply denoted $\|T\|_\alpha$; this is the standard construction when $X = Y$ or when equivalence of norms allows a unified choice. In contrast, mixed induced norms allow distinct norms $\|\cdot\|_\alpha$ and $\|\cdot\|_\beta$, which is useful in settings where the domain and codomain have incompatible natural norms, such as operators between $\ell^p$ and $\ell^q$ spaces with $p \neq q$. The general framework accommodates both cases while preserving the compatibility property.[](https://www.cis.upenn.edu/~cis5150/cis515-20-sl4.pdf) ### p-norm induced operators The p-norm induced operator norm, also known as the subordinate norm, arises when considering bounded linear operators $T$ acting between $\ell_p$ spaces, where the vector p-norm is given by $\|x\|_p = \left( \sum_{i=1}^\infty |x_i|^p \right)^{1/p}$ for $1 \leq p < \infty$ and $\|x\|_\infty = \sup_i |x_i|$.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) This norm measures the maximum amplification of the p-norm under $T$ and is defined as

|T|p = \sup{|x|_p = 1} |Tx|p = \sup{x \neq 0} \frac{|Tx|_p}{|x|_p}.

[](https://sites.math.washington.edu/~greenbau/Math_554/Course_Notes/ch1.3new.pdf) In finite dimensions, this applies to matrices acting on $\mathbb{R}^n$ or $\mathbb{C}^n$ equipped with the corresponding vector p-norms.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) Special cases of the p-norm induced operator norm correspond to particular values of p and admit explicit interpretations. For p=1, the induced norm $\|T\|_1$ equals the maximum absolute column sum of the matrix representation of $T$. For p=$\infty$, $\|T\|_\infty$ is the maximum absolute row sum. The case p=2 yields the [spectral norm](/page/spectral_norm), defined as the largest singular value of $T$, which is $\|T\|_2 = \sqrt{\rho(T^* T)}$, where $\rho$ denotes the [spectral radius](/page/spectral_radius) and $T^*$ is the adjoint.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) For an $n \times n$ matrix $A = (a_{ij})$ over the reals or complexes, the induced 1-norm and $\infty$-norm have closed-form expressions that facilitate computation: $\|A\|_1 = \max_{1 \leq j \leq n} \sum_{i=1}^n |a_{ij}|$ and $\|A\|_\infty = \max_{1 \leq i \leq n} \sum_{j=1}^n |a_{ij}|$.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) These formulas arise directly from the supremum definition, as the norms are attained by selecting unit vectors aligned with the coordinate axes that maximize the relevant sums.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) In contrast, the 2-norm requires more involved computation via singular value decomposition, though it remains the operator norm induced by the [Euclidean vector norm](/page/Euclidean_vector_norm).[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) A key duality property holds for these induced norms: for $1 < p < \infty$ with conjugate exponent q satisfying $1/p + 1/q = 1$, the p-induced norm of $T$ equals the q-induced norm of its adjoint $T^*$, i.e., $\|T\|_p = \|T^*\|_q$.[](https://sites.math.washington.edu/~greenbau/Math_554/Course_Notes/ch1.3new.pdf) This relation stems from the identification of the dual of $\ell_p$ with $\ell_q$ and the characterization of the operator norm via dual pairings.[](https://sites.math.washington.edu/~greenbau/Math_554/Course_Notes/ch1.3new.pdf) For p=2, the norm is self-dual since q=2.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) ## Non-Induced Norms ### Frobenius norm The Frobenius norm of an $ m \times n $ complex matrix $ A = (a_{ij}) $ is defined as

|A|F = \sqrt{\sum{i=1}^m \sum_{j=1}^n |a_{ij}|^2} = \sqrt{\trace(A^* A)},

where $ A^* $ denotes the conjugate transpose of $ A $, and $ \trace $ is the trace operator.[](https://mathworld.wolfram.com/FrobeniusNorm.html) This norm arises naturally from viewing the matrix as a vector in $ \mathbb{C}^{mn} $ equipped with the Euclidean norm, making it compatible with the standard inner product on the space of matrices given by $ \langle A, B \rangle_F = \trace(A^* B) $.[](https://mathworld.wolfram.com/FrobeniusNorm.html) In the more general setting of bounded linear operators on a separable [Hilbert space](/page/Hilbert_space) $ H $, the Frobenius norm extends to the [Hilbert-Schmidt norm](/page/Hilbert-Schmidt_norm) of an operator $ T: H \to H $, defined as

|T|{HS} = \sqrt{\sum{n=1}^\infty |T e_n|^2},

where $ \{e_n\}_{n=1}^\infty $ is any orthonormal basis of $ H $. This definition is independent of the choice of orthonormal basis, and the class of [Hilbert-Schmidt operators](/page/Hilbert-Schmidt_operators) forms a two-sided ideal in the algebra of bounded operators on $ H $, with the [Hilbert-Schmidt norm](/page/Hilbert-Schmidt_norm) inducing a Banach space structure. For finite-dimensional spaces, the [Hilbert-Schmidt norm](/page/Hilbert-Schmidt_norm) coincides with the [Frobenius norm](/page/Frobenius_norm) when $ T $ is represented by a matrix with respect to an orthonormal basis. A key property of the Frobenius norm is that it satisfies $ \|A\|_F \geq \|A\|_2 $ for any matrix $ A $, where $ \|A\|_2 $ is the [spectral norm](/page/spectral_norm) (the largest singular value of $ A $), with equality holding if and only if $ A $ has rank at most one.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) This inequality follows from the fact that the Frobenius norm is the Euclidean norm of the singular values of $ A $, while the spectral norm is their maximum. The Frobenius norm corresponds to the Schatten $ p $-norm for $ p=2 $, identifying it as the [Hilbert-Schmidt class](/page/Hilbert-Schmidt_class) of operators, which are compact and square-integrable in their integral kernel representations on $ L^2 $ spaces.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) The Frobenius norm finds applications in numerical linear algebra, particularly in least squares problems, where it measures perturbations in data matrices and solutions via condition numbers that bound relative errors under small disturbances.[](https://ccse.lbl.gov/Publications/sepp/leastSquares/LBNL-52434.pdf) For instance, in the sensitivity analysis of linear least squares, the Frobenius norm provides explicit bounds on the condition number, facilitating error estimates in computations.[](https://ccse.lbl.gov/Publications/sepp/leastSquares/LBNL-52434.pdf) It is also used in stability analysis of algorithms, as its computation requires only entrywise operations or a single matrix multiplication and trace evaluation, unlike the spectral norm which demands singular value decomposition.[](https://www.cis.upenn.edu/~cis5150/cis515-11-sl4.pdf) ### Schatten norms The Schatten *p*-norms generalize several familiar operator norms and for $ 1 \leq p < \infty $ are defined for compact operators on a separable Hilbert space $ H $. For a compact operator $ T \in B(H) $, the Schatten *p*-norm is

|T|p = \left( \sum{k=1}^\infty \sigma_k(T)^p \right)^{1/p},

where $ \sigma_1(T) \geq \sigma_2(T) \geq \cdots \geq 0 $ denote the singular values of $ T $. When $ p = \infty $, the Schatten $\infty$-norm is the usual operator (spectral) norm, $ \|T\|_\infty = \sup_k \sigma_k(T) $. The Schatten *p*-norms for $ 1 \leq p < \infty $ arise in the study of norm ideals of completely continuous (compact) operators and equip the corresponding spaces with a Banach space structure.[](https://link.springer.com/book/10.1007/978-3-642-87652-3)[](https://www.tntech.edu/cas/pdf/math/techreports/TR-2014-1.pdf) Particular cases of the Schatten *p*-norms recover well-known norms: for $ p=1 $, it yields the trace norm (or nuclear norm), $ \|T\|_1 = \sum_k \sigma_k(T) $, which measures the "total variation" of the singular values; for $ p=2 $, it is the Frobenius norm (as discussed in the prior section on non-induced norms). As $ p \to \infty $, $ \|T\|_p $ converges to the operator norm $ \|T\| $, bridging finite *p* cases to the supremum-based infinity norm. For 1 ≤ p < ∞, the Schatten classes $ S_p(H) = \{ T \in B(H) : \|T\|_p < \infty \} $ consist of compact operators on H, while $ S_\infty(H) = B(H) $. These classes form two-sided ideals in the C*-algebra $ B(H) $ of all bounded operators. These ideals satisfy strict inclusions $ S_p(H) \subset S_q(H) $ for $ 1 \leq p < q \leq \infty $, with continuous embeddings reflecting the decreasing summability requirements on the singular values as *p* increases.[](https://link.springer.com/book/10.1007/978-3-642-87652-3)[](https://www.tntech.edu/cas/pdf/math/techreports/TR-2014-1.pdf) Schatten norms play a prominent role in applications requiring control over singular value decay. In the regularization of ill-posed linear inverse problems, such as image deblurring or reconstruction, the Schatten *p*-norm of the Hessian operator serves as a convex regularizer that extends total variation methods by incorporating second-order information, thereby mitigating the staircase effect while preserving edges and enabling piecewise-linear solutions. In quantum information theory, the trace norm ($ p=1 $) underpins entanglement quantification; for instance, the logarithmic negativity of a bipartite state $ \rho $ on $ H_A \otimes H_B $ is defined as $ E_N(\rho) = \log_2 \|\rho^{T_A}\|_1 $, where $ T_A $ denotes partial transposition on subsystem $ A $, providing a computable upper bound on distillable entanglement that detects all entangled states. More broadly, Schatten norms facilitate measures of quantum correlations, including geometric quantum discord based on the Schatten 1-norm distance between states.[](https://www.skoltech.ru/app/data/uploads/sites/19/2016/07/tipJ2013.pdf)[](https://link.aps.org/doi/10.1103/PhysRevA.87.064101) ## Properties ### Basic properties The operator norm $\|\cdot\|$ on the space $B(X,Y)$ of bounded linear operators between normed vector spaces $X$ and $Y$ satisfies the standard axioms of a norm, endowing $B(X,Y)$ with the structure of a normed vector space.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) Positivity holds: for any $T \in B(X,Y)$, $\|T\| \geq 0$, and $\|T\| = 0$ if and only if $T$ is the zero operator. This follows directly from the definition $\|T\| = \sup_{\|x\| \leq 1} \|Tx\|$, as the supremum of non-negative quantities is non-negative, and it vanishes precisely when $Tx = 0$ for all $x \in X$.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) Homogeneity is also satisfied: for any scalar $c$ and $T \in B(X,Y)$, $\|cT\| = |c| \|T\|$. Indeed, $\|cT\| = \sup_{\|x\| \leq 1} \|c Tx\| = |c| \sup_{\|x\| \leq 1} \|Tx\| = |c| \|T\|$.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) The triangle inequality holds for any $S, T \in B(X,Y)$: $\|S + T\| \leq \|S\| + \|T\|$. To see this, note that for $\|x\| \leq 1$, $\| (S+T) x \| \leq \|Sx\| + \|Tx\| \leq \|S\| + \|T\|$, so taking the supremum yields the inequality. These three properties confirm that $\|\cdot\|$ defines a norm on $B(X,Y)$.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) The operator norm provides the minimal constant in the boundedness inequality: $\|Tx\| \leq \|T\| \|x\|$ for all $x \in X$, and $\|T\|$ is the smallest such constant. The inequality follows immediately from the definition by scaling, while the minimality arises because if $C < \|T\|$, there exists $x$ with $\|x\| \leq 1$ such that $\|Tx\| > C$, violating the bound.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) The supremum in the definition of $\|T\|$ is attained---that is, there exists $x$ with $\|x\| = 1$ such that $\|Tx\| = \|T\|$---if $X$ is finite-dimensional, or if $X$ is reflexive and $T$ is compact. In the finite-dimensional case, the unit sphere is compact and $x \mapsto \|Tx\|$ is continuous, so the maximum is achieved. For reflexive $X$ and compact $T$, attainment holds, as the image of the unit ball under a compact $T$ is relatively compact, and reflexivity ensures weak compactness sufficient for the norm to be realized.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) ### Multiplicativity The operator norm is submultiplicative: for bounded linear operators $S: Y \to Z$ and $T: X \to Y$ between normed spaces, $\|S \circ T\| \leq \|S\| \cdot \|T\|$.[](https://59clc.files.wordpress.com/2012/08/functional-analysis-_-rudin-2th.pdf) This follows directly from the definition. For any $x \in X$ with $\|x\| \leq 1$,

|(S T)x| = |S(Tx)| \leq |S| \cdot |Tx| \leq |S| \cdot |T| \cdot |x| \leq |S| \cdot |T|.

Taking the supremum over all such $x$ yields the inequality.[](https://59clc.files.wordpress.com/2012/08/functional-analysis-_-rudin-2th.pdf) Equality in submultiplicativity holds in specific cases, such as when one operator is an [isometry](/page/Isometry). If $T$ is an [isometry](/page/Isometry) ($\|Tx\| = \|x\|$ for all $x$), then $\|S T\| = \|S\|$, since the unit ball is preserved under $T$.[](https://www.princeton.edu/~aaa/Public/Teaching/ORF523/S17/ORF523_S17_Lec2_gh.pdf) For induced operator norms, the norm is compatible with adjoints: if $T: X \to Y$ is bounded, then $\|T^*\| = \|T\|$, where $T^*$ is the [adjoint](/page/Adjoint) operator.[](https://59clc.files.wordpress.com/2012/08/functional-analysis-_-rudin-2th.pdf) Submultiplicativity enables bounding powers of operators: for any bounded $T$, $\|T^n\| \leq \|T\|^n$ for positive integers $n$, by [induction](/page/Induction); this bound plays a key role in analyzing operator iterations and stability in [spectral theory](/page/Spectral_theory).[](https://59clc.files.wordpress.com/2012/08/functional-analysis-_-rudin-2th.pdf) ## Equivalent Characterizations ### Supremum definitions One equivalent characterization of the operator norm of a bounded linear [operator](/page/Operator) $ T: X \to Y $ between normed vector spaces $ X $ and $ Y $ utilizes the [dual space](/page/Dual_space) $ Y^* $ of continuous linear functionals on $ Y $. Specifically,

|T| = \sup \left{ |\phi(Tx)| : |x|X \leq 1, \ |\phi|{Y^*} \leq 1 \right},

where $ \phi \in Y^* $ and $ |\phi(Tx)| $ denotes the duality pairing between $ Y $ and $ Y^* $.[](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf) This expression arises because the norm on $ Y $ satisfies $ \|y\|_Y = \sup \{ |\phi(y)| : \|\phi\|_{Y^*} \leq 1 \} $ for any $ y \in Y $, so substituting $ y = Tx $ and taking the supremum over $ \|x\|_X \leq 1 $ yields the equivalence.[](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf) The double supremum can be interchanged without altering the value.[](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf) The standard definition $ \|T\| = \sup \{ \|Tx\|_Y : \|x\|_X \leq 1 \} $ can also be equivalently written using the closed [unit sphere](/page/Unit_sphere) instead of the closed unit [ball](/page/Ball), i.e., $ \|T\| = \sup \{ \|Tx\|_Y : \|x\|_X = 1 \} $. This holds because for any $ x $ with $ \|x\|_X < 1 $, [continuity](/page/Continuity) of $ T $ implies $ \|Tx\|_Y \leq \|T\| \cdot \|x\|_X < \|T\| $, so the supremum over the [ball](/page/Ball) is determined solely by values on the sphere.[](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf) Similarly, the supremum over the open unit [ball](/page/Ball) $ \{ x : \|x\|_X < 1 \} $ equals that over the closed unit [ball](/page/Ball), again due to the [continuity](/page/Continuity) of the [map](/page/Map) $ x \mapsto \|Tx\|_Y $, which ensures the values approach the [boundary](/page/Boundary) supremum arbitrarily closely.[](https://www.math.ucdavis.edu/~hunter/book/ch5.pdf) In infinite-dimensional spaces, these supremum characterizations highlight a key nuance: the operator norm need not be attained, meaning there may exist no $ x $ with $ \|x\|_X = 1 $ such that $ \|Tx\|_Y = \|T\| $ (or equivalently, no such pair $ (x, \phi) $ achieving the dual supremum). For instance, the backward [shift operator](/page/Shift_operator) $ B $ on $ \ell^2(\mathbb{N}) $, defined by $ B(e_1) = 0 $ and $ B(e_n) = e_{n-1} $ for $ n \geq 2 $ where $ \{e_n\} $ is the standard [orthonormal basis](/page/Orthonormal_basis), has $ \|B\| = 1 $ but satisfies $ \|Bx\|_2 < 1 = \|B\| $ for all $ x $ with $ \|x\|_2 = 1 $.[](https://www.math.uwaterloo.ca/~krdavids/FA/PM453Notes.pdf) This failure of attainment stems from the non-compactness of the unit ball in infinite dimensions, preventing the [continuous function](/page/Continuous_function) $ x \mapsto \|Tx\|_Y $ from achieving its maximum on [the sphere](/page/The_Sphere).[](https://www.math.ksu.edu/~nagy/func-an-2007-2008/bs-1.pdf) ### Finite-dimensional equivalents In finite-dimensional normed vector spaces $ V $ and $ W $, the operator norm of a bounded linear operator $ T: V \to W $ simplifies to $ \|T\| = \max_{x \neq 0} \frac{\|Tx\|}{\|x\|} $, where the maximum is attained due to the compactness of the closed unit sphere in $ V $ and the continuity of the map $ x \mapsto \|Tx\| $. This contrasts with the general supremum definition, as finite dimensionality ensures the norm is achieved at some nonzero vector $ x $, often related to a singular vector of $ T $.[](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/22bf2f774fb161776032491568148707_MIT18_102s21_lec2.pdf)[](https://www.ee.iitb.ac.in/~belur/uplod/golub-van-loan-matrix-computations-2012-edition-4th.pdf) For the induced 2-norm on matrices, the operator norm $ \|A\|_2 $ equals the square root of the largest eigenvalue of $ A^* A $, where $ A^* $ denotes the adjoint (conjugate transpose). This value is the maximum of the Rayleigh quotient $ \frac{x^* A^* A x}{x^* x} $ over unit vectors $ x $, attained at the corresponding eigenvector of $ A^* A $. Computationally, this requires finding the largest singular value via singular value decomposition or power iteration methods on $ A^* A $.[](https://www.ee.iitb.ac.in/~belur/uplod/golub-van-loan-matrix-computations-2012-edition-4th.pdf) In the context of $ \mathbb{R}^n $ or $ \mathbb{C}^n $ with standard p-norms, the induced operator norms of a [matrix](/page/Matrix) have simple explicit formulas. Specifically, $ \|A\|_1 = \max_j \|A e_j\|_1 = \max_j \sum_{i=1}^n |a_{ij}| $, the maximum absolute column [sum](/page/Sum) (achieved at [standard basis](/page/Standard_basis) vectors $ e_j $). For the infinity [norm](/page/Norm), $ \|A\|_\infty = \max_i \sum_{j=1}^n |a_{ij}| $, the maximum absolute row [sum](/page/Sum) (achieved at vectors with entries ±1 aligning the signs in the dominant row). These can be computed directly as the maximum absolute column [sum](/page/Sum) and row [sum](/page/Sum), respectively, or formulated as [linear programming](/page/Linear_programming) problems to optimize over the unit ball.[](https://www.ee.iitb.ac.in/~belur/uplod/golub-van-loan-matrix-computations-2012-edition-4th.pdf) ## Hilbert Space Operators ### Adjoint and norm equality In Hilbert spaces $H$ and $K$, the adjoint $T^*$ of a bounded linear operator $T: H \to K$ is the unique bounded linear operator $T^*: K \to H$ satisfying

\langle Tx, y \rangle_K = \langle x, T^* y \rangle_H

for all $x \in H$ and $y \in K$, where $\langle \cdot, \cdot \rangle_H$ and $\langle \cdot, \cdot \rangle_K$ denote the respective inner products.[](https://heil.math.gatech.edu/handouts/adjoint_hilbert.pdf) A fundamental property of the adjoint is that $\|T\| = \|T^*\|$. To see this, first note that the defining relation implies $\|T^* y\|_H = \sup_{\|x\|_H \leq 1} |\langle x, T^* y \rangle_H| = \sup_{\|x\|_H \leq 1} |\langle T x, y \rangle_K| \leq \|T\| \|y\|_K$, so $\|T^*\| \leq \|T\|$. For the reverse inequality, consider $\|T x\|_K^2 = \langle T^* T x, x \rangle_H \leq \|T^* T x\|_H \|x\|_H \leq \|T^*\| \|T x\|_K \|x\|_H$, which upon taking suprema over unit vectors yields $\|T\| \leq \|T^*\|$.[](https://heil.math.gatech.edu/handouts/adjoint_hilbert.pdf) The operator norm admits an equivalent characterization in terms of the inner product:

|T| = \sup \left{ |\langle T x, y \rangle_K| : |x|_H = |y|_K = 1 \right}.

This follows from the fact that $\|T x\|_K = \sup_{\|y\|_K = 1} |\langle T x, y \rangle_K|$ by the [Riesz representation theorem](/page/Riesz_representation_theorem), so taking suprema over unit $x$ gives the result; the reverse inequality holds by the Cauchy-Schwarz inequality.[](https://faculty.etsu.edu/gardnerr/Func/Beamer-Proofs/4-6.pdf) For [self-adjoint](/page/Self-adjoint) operators, where $T = T^*$ on a single [Hilbert space](/page/Hilbert_space) $H$, the norm simplifies further to

|T| = \sup_{|x|_H = 1} |\langle T x, x \rangle_H|.

Let $M = \sup_{\|x\|_H = 1} |\langle T x, x \rangle_H|$; clearly $M \leq \|T\|$ since $|\langle T x, x \rangle_H| \leq \|T x\|_H \|x\|_H = \|T\|$. For the opposite direction, note that the real part satisfies

\Re \langle T x, y \rangle_H = \frac{1}{4} \left( \langle T(x+y), x+y \rangle_H - \langle T(x-y), x-y \rangle_H \right) \leq M

for unit vectors $x, y$, with equality in the norm definition using the [polarization identity](/page/Polarization_identity). Thus, $\|T\| \leq M$.[](https://math.washington.edu/~hart/m556/notes2.pdf) ### Spectral considerations The spectral radius $\rho(T)$ of a bounded linear [operator](/page/Operator) $T$ on a [Hilbert space](/page/Hilbert_space) is defined as the supremum of $|\lambda|$ over all $\lambda$ in the [spectrum](/page/Spectrum) of $T$, and it satisfies $\rho(T) \leq \|T\|$ for the [operator norm](/page/Operator_norm) $\|T\|$.[](https://people.math.osu.edu/costin.9/7212-2020/FA1.pdf) This inequality arises because the powers of $T$ obey $\|T^n\| \leq \|T\|^n$, implying that the limit defining the [spectral radius](/page/Spectral_radius) cannot exceed the norm.[](https://people.math.osu.edu/costin.9/7212-2020/FA1.pdf) Gelfand's formula provides an explicit expression for the spectral radius: $\rho(T) = \lim_{n \to \infty} \|T^n\|^{1/n}$.[](https://people.math.osu.edu/costin.9/7212-2020/FA1.pdf) To sketch the proof, note that the operator norm is submultiplicative, so $\|T^{n+m}\| \leq \|T^n\| \|T^m\|$ for all positive integers $n, m$. This submultiplicativity implies that the sequence $a_n = \|T^n\|^{1/n}$ satisfies $a_{n+m} \leq (a_n a_m)^{n m / (n+m)}$ in a way that bounds the limsup by the infimum, establishing the limit's existence and equality to $\rho(T)$.[](https://people.math.osu.edu/costin.9/7212-2020/FA1.pdf) The inequality $\rho(T) \leq \|T\|$ then follows directly, as $\lim_{n \to \infty} \|T^n\|^{1/n} \leq \|T\|$ from the submultiplicativity.[](https://people.math.osu.edu/costin.9/7212-2020/FA1.pdf) For normal operators on Hilbert spaces, equality holds: $\|T\| = \rho(T) = \sup \{ |\lambda| : \lambda \in \sigma(T) \}$, where $\sigma(T)$ is the [spectrum](/page/Spectrum) of $T$.[](https://www.math.ksu.edu/~nagy/real-an/2-07-op-th.pdf) This result stems from the [spectral theorem](/page/Spectral_theorem) for [normal](/page/Normal) operators, which decomposes $T$ into a [multiplication](/page/Multiplication) operator by a bounded [measurable function](/page/Measurable_function) on $L^2(\mu)$, equating the norm to the essential supremum of the function's [modulus](/page/Modulus), which coincides with the [spectral radius](/page/Spectral_radius).[](https://www.math.ksu.edu/~nagy/real-an/2-07-op-th.pdf) This relation has applications in analyzing iterative processes; for instance, if $\|T\| < 1$, then $\|T^n\| \leq \|T\|^n \to 0$ as $n \to \infty$, implying $T^n \to 0$ in the operator norm and ensuring [convergence](/page/Convergence) of fixed-point iterations or [stability](/page/Stability) in dynamical systems governed by $T$.[](https://people.math.osu.edu/costin.9/7212-2020/FA1.pdf)

References

[1]
[PDF] Norms on Operators
operator norm on B(V) is submultiplicative (and L, G B(V) oL G B(V)). We want. to define functions of an operator L G B(V). We can compose L with itself, so we ...
[2]
[PDF] 4 Linear operators and linear functionals
In such cases we define the operator norm by. kTk = sup{kT(x)k : x ∈ V and ... See Theorem 3.9-4 in Introductory Functional Analysis with Applica-.<|control11|><|separator|>
[3]
[PDF] FUNCTIONAL ANALYSIS - ETH Zürich
Jun 8, 2017 · ... Functional Analysis. 57. 2.1 Uniform Boundedness ... operator norm. It also asserts that the sum of a Fredholm operator ...
[4]
[PDF] 18.102 S2021 Lecture 2. Bounded Linear Operators
Feb 18, 2021 · And in general, now that we've defined the operator norm, it gives us a bound of the form ... 18.102 / 18.1021 Introduction to Functional Analysis.<|control11|><|separator|>
[5]
[PDF] Elements of Functional Analysis - FA
Definition 1.2.3 (Operator norm). Let (V ,∥·∥V ) and (W,∥·∥W ) be normed vector spaces over the same scalar field and let T : V1. → V2 be a bounded.
[6]
[PDF] Operators on normed spaces - Matrix Editions
The next result says that for linear maps between normed spaces, bounded- ness and continuity are equivalent. Thus, a linear operator T : X → Y between ...
[7]
[PDF] History of Banach Spaces and Linear Operators
Jun 4, 2014 · ... Functional Analysis in Historical Perspective (1973), and. J. ... norm follow, and finally, completeness is required. The sole imperfection ...
[8]
Operator Norm -- from Wolfram MathWorld
The operator norm of a linear operator is the largest value by which stretches an element of , (1) It is necessary for and.
[9]
[PDF] Chapter 7 Vector Norms and Matrix Norms - UPenn CIS
The notion of subordinate norm can be slightly general- ized. Definition 7.8. If K = R or K = C, for any norm k k on Mm,n(K), and for any two norms k ka on ...
[10]
[PDF] Introduction to Functional Analysis
In functional analysis many different fields of mathematics come together. ... Now by definition of the operator norm‖̂𝑇‖ ≤ ‖𝑇‖ < ∞. In particular, ̂𝑇 ...
[11]
[PDF] Chapter 4 Vector Norms and Matrix Norms - UPenn CIS
The function A → A is called the subordinate matrix norm or operator norm induced by the norm . ... , and define the function R on Mn(R) by. A R. = sup.
[12]
Frobenius Norm -- from Wolfram MathWorld
The Frobenius norm, sometimes also called the Euclidean norm (a term unfortunately also used for the vector L^2 -norm), is matrix norm of an m×n matrix.
[13]
[PDF] Optimal Sensitivity Analysis of Linear Least Squares - CCSE
For full rank problems, Frobenius norm condition numbers are determined exactly, and spectral norm condition numbers are determined within a factor of square- ...
[14]
Norm Ideals of Completely Continuous Operators - SpringerLink
Free delivery 14-day returnsCompletely continuous operators on a Hilbert space or even on a Banach space have received considerable attention in the last fifty years.
[15]
[PDF] An Elementary Introduction to Schatten Classes
From the above lemma it follows that a compact linear operator on a Hilbert space either has only a finite number of non-zero eigenvalues or the sequence of ...
[16]
[PDF] Hessian Schatten-Norm Regularization for Linear Inverse Problems
These are used in a variational framework to derive regularized solu- tions of ill-posed linear inverse imaging problems.
[17]
Geometric quantum discord through the Schatten 1-norm
Jun 7, 2013 · Quantum discord is an information-theoretic measure of nonclassical correlations, initially proposed by Ollivier and Zurek [1] , which goes ...
[18]
[PDF] FUNCTIONAL ANALYSIS | Second Edition Walter Rudin
Functional analysis/Walter Rudin.-2nd ed. p. em. -(international series in ... operator norm by operators with finite-dimensional ranges. Hint: In a ...
[19]
[PDF] 1 Inner products and norms - Princeton University
Feb 9, 2017 · Notice that not all matrix norms are induced norms. An example is the Frobenius norm given above as ||I||∗ = 1 for any induced norm, but ||I||F ...Missing: domain codomain
[20]
[PDF] Banach Spaces - UC Davis Math
Theorem 5.33 Every finite-dimensional normed linear space is a Banach space. Proof. Suppose that (xk)∞ k=1 is a Cauchy sequence in a finite-dimensional normed.
[21]
[PDF] Functional Analysis - University of Waterloo
... not attained. Therefore α<f(a) for all a ∈ A. □. The following is immediate ... unilateral shift. Then S is isometric: kSxkp = kxkp. In particular, S ...
[22]
[PDF] Banach Spaces I: Normed Vector Spaces - KSU Math
As a special case of Proposition-Definition 6, we see that X∗ is naturally equipped with the norm, defined by kφk = sup |φ(x)| : x ∈ (X)1 . This norm is ...
[23]
[PDF] Golub and Van Loan - EE IIT Bombay
p-norm and Frobenius norm of a matrix (p. 71) dimension of a vector (p. 236) ... those of the matrix 1-norm or oo-norm. Fortunately, if the object is to ...
[24]
[PDF] functional analysis lecture notes: adjoints in hilbert spaces
Therefore, any non-self-adjoint operator provides a counterexample. For example, if H = Rn then any non-symmetric matrix A is a counterexample.
[25]
[PDF] Introduction to Functional Analysis
May 16, 2015 · T‖ by definition of the operator norm. So sup{‖Tx‖2 | ‖x‖ = 1} = ‖T‖2 ≤ ‖T∗T‖ and hence. ‖T∗T‖ = ‖T‖2. (). Introduction to Functional Analysis.
[26]
[PDF] spectral theory for compact self-adjoint operators
Suppose that T is a self-adjoint operator on H. Then. kTk = sup kxk=1. hT x, xi where kTk is the operator norm of T. Proof. Let M = supkxk=1. hT x, xi. , so M ...
[27]
[PDF] Class notes, Functional Analysis 7212 - OSU Math
Apr 1, 2019 · In this section all operators are bounded, where the norm, or operator norm, of an operator B is defined as. kBk = sup u,kuk=1. kBuk = sup x6 ...
[28]
[PDF] 7. Operator Theory on Hilbert spaces - KSU Math
For every self-adjoint operator T ∈ B(H), one has the equality. radH(T) = kTk. Proof. It T = 0, there is nothing to prove, so without any loss of generality.