Definition of capitalism Does the discipline of economics have an accepted rigorous definition of capitalism?
 I.e. not something like "some economists would quote Hayek and say that so and so constitutes capitalism" but an if and only if definition that most economists would agree on.
The dictionary definitions of capitalism (currently the most upvoted answer) have been critiqued as inadequate, in the sense that they are too broad, by Geoffrey Hodgson in his treatise, Conceptualising Capitalism (University of Chicago Press, 2015).


  Definitions proliferate. Most dictionaries stress private
  ownership and markets; many add the profit motive. The entry in The
   New Encyclopaedia Britannica (1998, 2:831) is typical: “Capitalism, also
  called free market economy, or free enterprise economy: economic system
  . . . in which most of the means of production are privately owned
  and production is guided and income distributed largely through the operation
  of markets.”
  
  But, if we consider systems based largely on private ownership, the
  profit motive, and markets, we can find many examples before the medieval
  era. [...] trade has existed for tens of
  thousands of years, markets may have appeared as long as five millennia
  ago in China, and they were evident in several locations in the eastern
  Mediterranean and the Middle East around 600 BC. Private ownership,
  the pecuniary calculus, and the profit motive have similarly played
  a prominent role in leading economic regions for thousands of years.
  Hence, prominent dictionary definitions of capitalism make it a system
  that has lasted for multiple millennia. If we adopt such definitions, then
  we need to find another word to describe the institutional structures that
  were consolidated in the eighteenth century and gave rise to spectacular
  economic growth and ongoing technological innovation.


And he does have his own definition (which I will not try to argue that is mainstream); its premise is that: 


  Capitalism [...] requires special forms of state that cannot confiscate property arbitrarily at will, that are effectively restrained by laws, that have internal
  checks and balances, and that are faced with countervailing (democratic
  or other) powers that help protect a relatively autonomous legal system.
  Such states are necessary to legitimate and protect property rights and
  to enforce contracts. They required peculiar circumstances and a long
  time to evolve. Foreshadowed in the Italian city- states, they did not appear
  on a national scale until the seventeenth century, in Britain and the
  Netherlands. [...]
  
  I argue that, while markets are central to capitalism, capitalism is not
  simply a market system: unavoidably it contains different subsystems of
  governance, production, distribution, and exchange. Furthermore, capitalism
  cannot in principle have markets for everything or bring everything
  within the orbit of commodity exchange.
  In particular, under capitalism there can never be a complete set of
  markets for future labor power. For there to be full futures markets for
  labor, all workers must enter into contracts for their expected working
  life. This would be tantamount to voluntary bondage, limiting the freedom
  of workers to quit their employment. Paradoxically, pushing markets
  to their limits would mean the return of slavery for the workforce.


He goes on to semi-formalize a list of tenets that constitute capitalism first according to [his interpretation of] Marx and then similarly for Schumpeter


  M[arx]-capitalism
  
  
  A legal system supporting widespread individual rights and liberties to own,
  buy, and sell private property
  Widespread commodity exchange and markets involving money
  Widespread private ownership of the means of production by firms producing
  goods or services for sale in the pursuit of profit
  Much of production organized separately and apart from the home and
  family
  Widespread wage labor and employment contracts
  
  
  Most dictionary and encyclopedia definitions emphasize conditions
  (1), (2), and (3), albeit often without emphasis on the legal system. More
  than Marx, the notion of M-capitalism emphasizes the legal system and
  individual rights. Regarding condition (4), the separation of much production
  from the family and the domestic sphere was emphasized by Weber
  (1968) and is implicit in Marx’s writings. Weber rightly argued that
  this separation was important to subject production to systems of rational
  accounting and pecuniary motivation. Marx emphasized condition
  (5) because it flows from the nature of capitalism as generalized commodity
  production and encapsulates the driving antagonism between
  the employing class and the working class. [...]
  
  A problem [...] is that the addition of
  condition (5), widespread wage labor, to the preceding four does not effectively
  demarcate modern industrial capitalism from the four preceding
  centuries when wage labor was widespread in rural England.
  M-capitalism, as specified above, gives us a better-than-average definition,
  but not one that is sufficiently sharp in terms of historical specificity.
  
  Further problems of historical demarcation arise with the first four
  conditions. As noted in section 4.4 above, the idea that secure property
  rights were first established in England in the seventeenth century is a
  myth. Furthermore, money, markets, and private ownership have existed
  for millennia. The fourth condition is helpful but hardly sufficient as an
  explanation of massive expansion of capitalism around 1800. The five
  conditions are inadequate to demarcate historically the explosion of capitalist
  productivity even if they are necessary for that system.


As for S[chumpeter]-capitalism, he argues that it contains all of the above, except for #5, but that additionally it has


  
  A developed financial system with banking institutions, the widespread use
  of credit with property as collateral, and the selling of debt
  
  
  The addition of condition (6) takes seriously the monetary definition
  of capital and institutions promoting collateralization and the salability
  of debt. Capitalism is thus marked by “the predominance of ‘capital’”
  (Sombart 1930, 196). This definition points to the development of institutions
  of clearer historical specificity. According to this definition, the
  emergence of capitalism in England is marked by developments in financial
  institutions in the eighteenth century.
  
  Note that condition (1) in the definitions of M-capitalism and
  S-capitalism rules out (at least widespread) slavery. Consequently, both
  M-capitalism and S-capitalism involve missing futures markets for labor.
  With S-capitalism, there may be self-employment or worker cooperatives
  but not extensive slave labor. M-capitalism rules out an economy
  with predominant self- employment or worker cooperatives.


As you can probably guess, by synthesis, he argues that the definition of capitalism should include the superset of 1-6 from both M- and S-capitalism (which already have a lot in common).


  But there is a cost to the exclusion of condition (5) from this definition. [...] the advantage of retaining
  the employment contract in the definition of capitalism is that it demarcates
  the system from economies involving self-employed producers or
  worker cooperatives. Furthermore, the introduction of wage labor gave
  important incentives for labor-saving innovations and led to increases in
  productivity. Consequently, the definition of capitalism that is favored
  here— and described simply as capitalism, without any prefix— involves
  all six of the above conditions. [...] 
  
  Conditions
  (5) and (6) serve as fuzzy historical bookends, together demarcating
  capitalism from preceding systems and possible successors. [...]
  
  Explicit reference to social class is also absent from the definition offered
  above. In his review of different definitions of capitalism, Michael
  Merrill (1995) argued that capitalism is much more than simply a market
  economy. He was right about that. But he wanted to define capitalism as
  “a market economy ruled by, or in the interests of, capitalists” (322). A
  problem [...] is that the basis of class itself has to
  be defined, and Marx and Engels were obliged to use legal terms such as
  ownership to do so. Being a capitalist or a worker is not a congenital attribute:
  it is the occupation of a social role that is determined in part by
  legal relations. Class-related legal terms appear in all points above, except
  condition (4).
  
  Marxists have abused class-centered definitions of capitalism. They
  see class struggle, where the destiny of one class is to seize control of the system from another. But, notwithstanding the power of capitalists over
  workers, capitalists do not control capitalism: it is more the other way
  round. Such large and complex systems cannot be fully controlled, as if
  from a single chamber of power.


And while he disses that idea that capitalists control capitalism itself as a caricature, he nevertheless argues that it is an integrated economic and political system (and in this he is not that original, as he acknowledges):


  The central role of the state within capitalism means that we must
  address politics as well as economics. [...] Douglass C. North, John J. Wallis,
  and Barry R. Weingast (2009, 269) argued: “The seeming independence
  of the economic and political systems on the surface is apparent, not
  real. In fact, these systems are deeply intertwined.” I also concur with
  Bruce R. Scott (2009, 4) in his claim that capitalism is both “a political
  phenomenon” and “an economic one” and that “specifically it requires
  the visible hands of political actors exercising power through political
  institutions.” Capitalism always involves legal and political institutions:
  pure “anarchocapitalism” is an unrealizable fantasy.


And since the Austrians (Hayek actually) got mentioned in the question, Hodgson has this to say about their approach to defining capitalism:


  The Austrian economists had
  little to say about the sphere of production—conditions (4) and (5)—
  including the employment contract. Although they mentioned money
  many times, they paid insufficient attention to the use of property as collateral
  or to the selling of debt—condition (6). Their theory of money
  downplays the vital constitutive role of the state. At best, they stressed
  three of the six conditions only. The first three conditions are insufficient to identify the institutions of the capitalist system that were responsible
  for the explosive production of wealth in the last three hundred
  years.


While this may not be any sort of consensus statement, it is at least a comparative analysis of various definitions (from fairly prominent sources).