A map of topological spectra is a compatible family of based maps of the level spaces: $u_{j+1}{\sigma }_j^D={\sigma }_j^E(1\wedge u_j)$. It is distinct from its class in the stable homotopy category.

An Omega-spectrum has structure adjoints $E_j\to \Omega E_{j+1}$ that are weak homotopy equivalences. For such a model the level spaces represent the degrees of its represented cohomology theory.

A cell spectrum is obtained by attaching stable cells, with successive cofibers of spectra wedges of suspensions of spectra of the sphere spectrum. Cellular models provide the cofibrant objects needed to compute derived maps of topological spectra. Stable cells may occur in arbitrary integer degrees.

For a based space $X$, its suspension spectrum has level $j$ equal to $S^j\wedge X$. Its derived adjunction with the infinite loop space is $[{\Sigma }^{\infty }X,E{]}_{\mathrm{st}}=[X,{\Omega }^{\infty }E{]}_{\ast }$.

Suspension of spectra shifts the stable degree. Unlike reduced suspension of based spaces, it is invertible in the stable homotopy category, with inverse denoted ${\Sigma }^{-1}$.

The $k$th stable homotopy group of $E$ is ${\pi }_{k}E=[{\Sigma }^k\mathbb{S},E{]}_{\mathrm{st}}$, equivalently ${\mathrm{colim}}_j{\pi }_{k+j}{E}_j$ in a sequential model.

An indexed prespectrum over $U$ consists of based spaces $E(V)$ indexed by finite-dimensional $V\subset U$, with unital associative maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for inclusions $V\subset W$. The orthogonal differences make iterated suspension coordinates compose consistently.

A ring spectrum is a unital associative monoid object of the stable homotopy category under the smash product of spectra. It has $\mu :E\wedge E\to E$ and $\eta :\mathbb{S}\to E$, satisfying the associative and two unit identities. Structured ring spectra additionally specify coherent point-set or higher homotopies; a ring object alone does not assert those enhancements.

A connective spectrum has ${\pi }_{q}E=0$ for $q<0$. This makes the homological Atiyah-Hirzebruch spectral sequence first quadrant for a based space, and prevents negative coefficient degrees from affecting its first nonzero represented homology theory.

In a unital Banach algebra, for $\lambda \ne 0$ the existence of $R=(\lambda 1-ab{)}^{-1}$ gives $(\lambda 1-ba{)}^{-1}={\lambda }^{-1}(1+bRa)$, as verified by multiplying on either side. Interchanging $a,b$ proves the converse. Zero may differ: for the unilateral shift operator $S$ on ${\ell }^2({\mathbb{N}}_0)$, $S^{\ast }S=1$ has spectrum $\{1\}$ and $S{S}^{\ast }=1-P_0$ has spectrum $\{0,1\}$.

An element of a complex unital Banach algebra is quasinilpotent if its spectral radius is zero, equivalently its spectrum of an element is $\{0\}$. A nilpotent element is quasinilpotent, but the converse need not hold. The Volterra convolution algebra on a finite interval gives powers which never vanish, yet have factorially decreasing norms, making the inverse series for $\lambda e-x$ converge for every $\lambda \ne 0$.

The identity $(x+\alpha e)-\lambda e=x-(\lambda -\alpha )e$ proves the formula directly from invertibility. In a nonzero complex unital Banach algebra, the nonemptiness of the Banach-algebra spectrum implies that if $x$ is a quasinilpotent element, then $\rho (x+e)=1$. Consequently every nonscalar element cannot simultaneously be quasinilpotent: adding the identity preserves nonscalarity while translating its spectrum.

For a closed unital subalgebra $B$ of a unital Banach algebra $A$ and $x\in B$, the smaller-algebra algebra resolvent set is relatively open and closed in the larger one. Openness follows from the Neumann series; closedness follows from continuity of inversion in $A$ and norm closedness of $B$. Every larger-algebra resolvent component is therefore either retained whole or absorbed into the smaller-algebra Banach algebra spectrum. The unbounded component is always retained, since inverses at large modulus are norm-convergent series in $x$.

The algebra resolvent set consists of those scalars for which $\lambda 1-x$ is invertible in the algebra. It is open by the Neumann series, and the inverse-valued map is the resolvent of an element. A closed subalgebra can have a smaller algebra resolvent set because an ambient inverse need not lie in it.

Spectrophotometry measures received light as a function of wavelength, accounting for the instrument's wavelength-dependent response.

The Biot strain tensor uses $f(\lambda )=\lambda -1$, so $E^{(1)}=U-I$. Its principal values are extension ratios minus one. It is a spectral strain measure, since $f(1)=0$, $f^{\prime }(1)=1$ and $f^{\prime }>0$. Its reference-volume work conjugate is the symmetric Biot stress.

In vacuum, $\nu =c/\lambda$, where $c$ is the speed of light. Increasing wavelength decreases frequency, so the ordinary signed differentials satisfy $d\nu =-c{\lambda }^{-2}d\lambda$. A spectral slice's positive contribution to radiance must be unchanged. The change of variables formula therefore gives the absolute Jacobian determinantThese formulas use $L_{\lambda }$ per metre and $L_{\nu }$ per hertz. The factor $c/{\nu }^2$ has units $\mathrm{m}\mathrm{s}=\mathrm{m}\mathrm{H}{\mathrm{z}}^{-1}$, which converts the former density into the latter. If ${\ell }_{\mathrm{nm}}$ denotes the numerical density per nanometre in $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$, its per-metre value is $10^9{\ell }_{\mathrm{nm}}$. Conversely, the per-nanometre value is $10^{-9}$ times the per-metre value. Likewise, a density per terahertz is $10^{12}$ times its per-hertz value because $1\mathrm{THz}=10^{12}\mathrm{Hz}$.

For a concrete stipulated spectrum, let $L_{\lambda }=C=10^9\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}{\mathrm{m}}^{-1}$ between ${\lambda }_1=500\mathrm{nm}$ and ${\lambda }_2=600\mathrm{nm}$, and zero outside. This is exactly $1\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$ throughout the band. Using the exact International System of Units value $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, the corresponding frequency bounds areThe transformed spectral radiance is $L_{\nu }=Cc/{\nu }^2$ on this interval. At ${\nu }_1$ it is approximately $1.200830743\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$, and at ${\nu }_2$ it is $8.339102380\times 10^{-13}$ in the same units. These are respectively $1.200830743$ and $0.833910238$ per terahertz. Thus a flat wavelength density becomes a decreasing frequency density; relabelling the horizontal axis alone would miss both the shape and the units. ^[ref].

The equality of band radiances has a short exact proof:The reversed bounds and the Jacobian determinant preserve the same spectral measure. Each half-band, $500$–$550\mathrm{nm}$ and $550$–$600\mathrm{nm}$, carries $50\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, although the corresponding frequency widths are about $54.508$ and $45.423\mathrm{THz}$.

A spectrum's peak is consequently a statement about a particular density and coordinate. The frequency and wavelength peaks of Planck's law differ for precisely this reason. Integrated radiance over a specified physical band supplies a comparison that survives the change of variables.

A Voigt profile is the convolution of a Gaussian distribution and a Cauchy distribution. For an absorption line, thermal Doppler broadening contributes the Gaussian core while finite excited-state lifetime contributes the Lorentzian wings. Instrumental broadening and unresolved motions must also be modeled when inferring column density and temperature.

Collisions and fluctuating neighboring-particle electric fields alter atomic energy levels and coherence times, broadening a spectral line. In a simple impact treatment the coherence-decay rate gives a Lorentzian width of order $\Gamma /(2\pi )$; the precise definition of full or half width must be stated. Unlike Doppler broadening, this depends primarily on collision rates and perturber density. In stellar opacity calculations it alters line overlap and the low-opacity windows entering the Rosseland mean opacity.

For a nonnegative elliptic operator $L$ on a closed manifold, sum its positive eigenvalues with multiplicity. The defining series converges in a right half-plane. If $n_0=\dim \ker L$ and $K(t)=\mathrm{Tr}(e^{-tL})$ is the heat trace, thenSubtracting finitely many terms of the short-time heat kernel expansion gives a meromorphic continuation. Zero eigenvalues must be excluded: assigning them a nonexistent inverse power does not define the spectral zeta function.